Machine Learning for Potential Energy Surfaces: A Comprehensive Guide for Computational Researchers and Drug Developers

Abstract

This article provides a comprehensive overview of how Machine Learning (ML) is revolutionizing the exploration of Potential Energy Surfaces (PES), a cornerstone for understanding molecular interactions and dynamics. Tailored for researchers and drug development professionals, we cover the foundational principles of ML-driven PES, from automated frameworks that streamline data generation to advanced kernel and neural network models. The article delves into practical methodologies, including Δ-machine learning for cost-effective high-accuracy surfaces, and addresses critical challenges like data quality and model generalizability across different chemical spaces. Finally, we present rigorous validation protocols and comparative analyses of state-of-the-art models, highlighting their transformative implications for accelerating drug discovery, from target identification to formulation.

Demystifying ML-Driven Potential Energy Surfaces: From Core Concepts to Automated Exploration

The PES Bottleneck in Traditional Computational Chemistry and Materials Science

The precise calculation of Potential Energy Surfaces (PES) represents one of the most fundamental challenges in computational chemistry and materials science. These multidimensional surfaces dictate atomic interactions, molecular reactivity, and material properties, yet their accurate determination requires computationally expensive quantum mechanical calculations that create a significant bottleneck for research progress. For polyatomic systems with multiple degrees of freedom, high-level ab initio calculations with electron correlation are exceptionally demanding, making comprehensive PES exploration practically impossible for many systems of scientific and industrial interest [1]. This bottleneck fundamentally limits our ability to understand reaction kinetics, predict material behavior, and accelerate drug discovery processes where molecular interactions are paramount.

The core challenge lies in the exponential scaling of computational cost with system size and accuracy requirements. Traditional electronic structure methods, while accurate, become prohibitively expensive as molecular complexity increases, forcing researchers to compromise either on system size or on the accuracy of their calculations. This limitation has stimulated the development of innovative computational approaches that combine theoretical chemistry with machine learning to overcome the PES bottleneck, opening new frontiers in atomistic simulation [1] [2].

The Computational Bottleneck: Scope and Scale of the Problem

Quantitative Demands for Accurate PES Construction

Table 1: Computational Requirements for High-Accuracy PES Development in Representative Chemical Systems

System	Method	Number of Energy Points	Accuracy Target	Key Challenges
H + CH4 Reaction	PIP-NN [1]	~63,000 ab initio points	0.12 kcal mol-1 (42 cm-1)	Hydrogen abstraction dynamics, tunneling effects
H + CH4 Reaction	Δ-ML [1]	Large LL set + small HL correction	Chemical accuracy (~1 kcal mol-1)	Efficient sampling, transferability
Titanium-Oxygen System	GAP-RSS [2]	Thousands of DFT single points	~0.01 eV/atom	Multiple stoichiometries, polymorph diversity
Small Molecules (≤15 atoms)	VCI [3]	High-order PES expansion	1-5 cm-1 for fundamentals	Convergence of vibrational calculations

The demands for constructing accurate PESs vary significantly based on the system complexity and desired application. For kinetic and dynamic studies of chemical reactions, such as the H + CH₄ hydrogen abstraction reaction, thousands of high-level ab initio calculations are typically required to achieve chemical accuracy (approximately 1 kcal mol^-1) [1]. For materials systems like titanium-oxygen compounds with multiple polymorphs and stoichiometries, the configurational space expands dramatically, requiring sophisticated sampling strategies [2]. Meanwhile, for spectroscopic applications of small molecules, the emphasis shifts to extremely precise local PES representations around minima to achieve wavenumber accuracy better than 5 cm^-1 for fundamental transitions [3].

Traditional Approaches and Their Limitations

Traditional approaches to PES construction face fundamental limitations in both efficiency and applicability. The n-mode expansion method, which represents the PES through a series of increasingly complex many-body terms, suffers from the "curse of dimensionality" - the exponential increase in required calculations as both system size and expansion order increase [3]. For example, a quartic force field (QFF) expansion provides a reasonable balance between accuracy and computational cost for some systems, but fails dramatically for molecules with significant anharmonicity or multiple minima [3].

Table 2: Accuracy Comparison of PES Expansion Truncation for Vibrational Frequencies (cm^-1)

Molecule	VPT2	VCI(QFF)	VCI(2D)	VCI(3D)	VCI(4D)
H2CO	1.5 (3.1)	6.2 (12.3)	13.1 (51.5)	2.4 (7.4)	1.4 (3.2)
CH2F2	1.8 (5.3)	5.4 (16.2)	11.1 (80.8)	2.0 (8.4)	1.5 (3.8)
C2H4	2.9 (9.0)	9.9 (26.2)	10.7 (27.4)	10.6 (34.0)	2.7 (5.9)
NH2CHO	28.7 (173.5)	192.1 (474.1)	30.2 (125.2)	22.8 (70.1)	3.2 (9.4)

Note: Values represent mean absolute deviation (maximum deviation in parentheses) from experimental fundamental frequencies [3]

As shown in Table 2, the truncation order of the PES expansion dramatically impacts the accuracy of subsequent vibrational spectrum calculations. While second-order vibrational perturbation theory (VPT2) performs reasonably well for many systems, it fails catastrophically for molecules like formamide (NH₂CHO). Similarly, variational calculations based on quartic force fields (VCI(QFF)) show unacceptably large errors. Only high-order n-mode expansions (VCI(4D)) consistently achieve the required accuracy across diverse molecular systems, but at tremendous computational cost [3].

Machine Learning Solutions to the PES Bottleneck

Δ-Machine Learning: A Hybrid Approach

Delta-machine learning (Δ-ML) has emerged as a highly cost-effective strategy for developing high-level PESs by leveraging the complementary strengths of low-level and high-level computational methods [4] [1]. The fundamental equation underlying this approach is:

[ V{i}^{HL} = V{i}^{LL} + \Delta V_{i}^{HL-LL} ]

where the superscript ( i ) refers to the ( i^{th} ) geometric configuration, ( HL ) denotes high-level, and ( LL ) denotes low-level [1]. The power of this method lies in the fact that the correction term, ( \Delta V^{HL-LL} ), is a slowly varying function of atomic coordinates and can therefore be machine-learned from a relatively small number of judiciously chosen high-level data points.

Diagram 1: Δ-Machine Learning Workflow for PES Development. This schematic illustrates the hybrid approach that combines extensive low-level data with targeted high-level corrections to efficiently generate accurate potential energy surfaces [1].

In the Δ-ML approach applied to the H + CH₄ reaction, the PES-2008 analytical surface served as the low-level reference, while high-level energies were obtained from a permutation invariant polynomial neural network (PIP-NN) surface [1]. This strategy successfully reproduced kinetics and dynamics information of the high-level surface with significantly reduced computational cost, demonstrating its efficiency in describing multidimensional polyatomic systems.

Automated Framework: The autoplex Approach

The autoplex framework represents a different machine learning strategy focused on automating the exploration and fitting of potential-energy surfaces [2]. This approach uses iterative, data-driven random structure searching (RSS) to efficiently explore configurational space while gradually improving machine-learned interatomic potentials (MLIPs). The key innovation lies in using gradually improved potential models to drive searches without relying on any first-principles relaxations or pre-existing force fields, requiring only density functional theory (DFT) single-point evaluations [2].

Diagram 2: Automated PES Exploration with autoplex. This workflow illustrates the iterative process of random structure searching guided by machine-learned interatomic potentials, which enables efficient exploration of complex energy landscapes [2].

The performance of autoplex has been demonstrated across systems of increasing complexity, from elemental silicon to the full binary titanium-oxygen system. For silicon, the highly symmetric diamond-type and β-tin-type structures were accurately described with approximately 500 DFT single-point evaluations, while the more complex oS24 allotrope required a few thousand evaluations to achieve the target accuracy of 0.01 eV/atom [2]. This progressive approach to system complexity highlights the framework's capability to handle diverse materials challenges.

Experimental Protocols and Methodologies

Detailed Δ-ML Protocol for Reaction PES

The application of Δ-machine learning to chemical reaction PES development follows a structured protocol:

• Low-Level PES Selection: Choose an appropriate analytical or computational low-level PES that provides reasonable coverage of the configurational space. For the H + CH₄ system, the PES-2008 surface based on valence-bond molecular mechanics (VB-MM) was employed [1].

• Configuration Sampling: Extract a large set of molecular configurations (typically thousands to tens of thousands) from the low-level PES, ensuring adequate coverage of reactants, products, transition states, and relevant asymptotic regions [1].

• High-Level Reference Selection: Identify a suitable high-level reference method. In the H + CH₄ case, the PIP-NN surface fitted to UCCSD(T)-F12a/AVTZ calculations with ~63,000 points served as the high-level benchmark [1].

• Strategic Subset Selection: Choose a judicious subset of configurations (typically much smaller than the full set) for high-level evaluation. This selection should capture the essential physics and chemistry of the system while minimizing computational cost.

• Machine Learning Correction: Train a machine learning model (neural networks, Gaussian process regression, etc.) on the difference between high-level and low-level energies (( \Delta V^{HL-LL} )) for the subset of configurations.

• Validation: Perform comprehensive kinetic and dynamic validation. For the H + CH₄ system, this included variational transition state theory with multidimensional tunneling corrections and quasiclassical trajectory calculations for the deuterated reaction H + CD₄ [1].

Automated RSS-MLIP Protocol for Materials

The automated random structure searching combined with machine-learned interatomic potentials follows this methodology:

• Initialization: Define the chemical system (elements, composition ranges) and generate an initial set of random structures.

• DFT Parameter Setup: Establish consistent DFT parameters (exchange-correlation functional, basis set/pseudopotentials, convergence criteria) for all single-point calculations.

• Iterative RSS-MLIP Cycle:

  • Relax structures using the current MLIP (initially, if no MLIP exists, use very short DFT relaxations)
  • Select diverse structures for DFT single-point calculations
  • Compute DFT energies and forces for selected structures
  • Add results to the training dataset
  • Retrain MLIP on the expanded dataset
  • Assess convergence using root mean square error (RMSE) between predicted and reference energies

• Performance Evaluation: Test the final MLIP on known polymorphs and compositions not included in the training set, evaluating both energy and force accuracies [2].

• Production Simulations: Employ the validated MLIP for large-scale molecular dynamics, phase stability analysis, or property calculations.

The Scientist's Toolkit: Essential Research Reagents

Table 3: Key Computational Tools for Machine Learning-Enhanced PES Exploration

Tool/Resource	Type	Primary Function	Application Examples
Permutation Invariant Polynomial Neural Networks (PIP-NN)	Machine Learning Architecture	Constructs PESs invariant to atomic permutation	H + CH4 reaction surface [1]
Gaussian Approximation Potentials (GAP)	Machine Learning Framework	Data-efficient interatomic potentials for materials	Titanium-oxygen system exploration [2]
autoplex	Automated Workflow Software	Integrates RSS with MLIP fitting	High-throughput materials screening [2]
Δ-ML Methodology	Computational Strategy	Combines LL and HL calculations for efficient PES	Reaction kinetics and dynamics [4] [1]
Stochastic Hyperspace Embedding and Projection (SHEAP)	Visualization Algorithm	Dimensionality reduction for energy landscape visualization	Mapping funnels in Lennard-Jones clusters [5]
n-mode Expansion	Mathematical Representation	Expands PES as sum of many-body terms	Vibrational spectrum calculations [3]
Random Structure Searching (RSS)	Sampling Method	Explores configurational space efficiently	Crystal structure prediction [2]
U0126-EtOH	U0126-EtOH, CAS:1173097-76-1, MF:C20H22N6OS2, MW:426.6 g/mol	Chemical Reagent	Bench Chemicals
S-Propargylcysteine	S-Propargylcysteine, CAS:3262-64-4, MF:C6H9NO2S, MW:159.21 g/mol	Chemical Reagent	Bench Chemicals

The tools summarized in Table 3 represent the essential computational "reagents" required for modern PES exploration. These resources enable researchers to overcome the traditional bottlenecks through automated workflows, efficient machine learning architectures, and sophisticated sampling strategies. The field has progressed from hand-crafted models tailored for specific systems to automated frameworks capable of exploring complex multi-element systems with minimal user intervention [2].

The PES bottleneck in traditional computational chemistry and materials science is being systematically addressed through innovative machine learning approaches that combine physical principles with data-driven methodologies. Δ-machine learning provides a cost-effective pathway to high-accuracy surfaces by leveraging the complementary strengths of low-level and high-level computational methods [1]. Simultaneously, automated frameworks like autoplex demonstrate how random structure searching combined with iterative MLIP improvement can efficiently explore complex materials systems [2].

These advances are transforming computational modeling from a specialized, labor-intensive activity to a more automated, accessible tool for researchers across chemistry, materials science, and drug discovery. As machine learning methodologies continue to mature and integrate more deeply with physical theories, we anticipate further acceleration in PES exploration capabilities, ultimately enabling the realistic simulation of increasingly complex systems with quantum-mechanical accuracy.

How Machine Learning Acts as a Surrogate for Quantum-Mechanical Calculations

Quantum-mechanical (QM) calculations, particularly those based on density-functional theory (DFT), provide the foundation for modern computational chemistry and materials science, enabling researchers to predict chemical properties, reaction pathways, and material behaviors with high accuracy [2]. However, this accuracy comes at a significant computational cost that makes direct QM calculations prohibitive for large molecular systems or extended time-scale simulations [6]. The computational expense grows rapidly with system size, rendering routine studies of complex biological molecules or materials with thousands of atoms practically infeasible. This fundamental limitation has driven the development of machine learning (ML) surrogates that can learn the intricate mapping between chemical structure and QM-derived properties from reference data, then make accurate predictions at a fraction of the computational cost [2] [6].

The core premise of ML-as-surrogate lies in replacing the explicit numerical solution of the electronic Schrödinger equation with a trained statistical model that captures the underlying physical relationships. By learning from a carefully generated training set of QM calculations, these models can achieve what Anima Anandkumar describes as "transferability to larger molecules" – the ability to make accurate predictions for molecular systems significantly larger than those present in the training data [6]. This paradigm shift enables researchers to perform quantum chemistry calculations up to 1,000 times faster than previously possible, transforming workflows that previously took days into interactive computing sessions [6].

Core Methodological Approaches

Machine-Learned Interatomic Potentials (MLIPs)

Machine-learned interatomic potentials have emerged as the method of choice for large-scale, quantum-mechanically accurate atomistic simulations [2]. MLIPs are trained on quantum-mechanical reference data—typically derived from DFT—using methods ranging from linear fits and Gaussian process regression to neural-network architectures [2]. The Gaussian approximation potential (GAP) framework, which leverages the data efficiency of Gaussian process regression, has proven particularly successful for constructing MLIPs through automated exploration of potential-energy surfaces (PES) [2].

The fundamental operation of an MLIP involves learning the relationship between atomic configuration and potential energy, such that the total energy of a system is expressed as a sum of local atomic environments. This approach enables the ML model to make predictions for structures not explicitly included in the training set, generalizing across chemical space. As Behler and Parrinello established in their seminal work, this decomposition allows for the creation of potentials that remain computationally efficient while maintaining quantum-mechanical accuracy [2].

Delta-Machine Learning (Δ-ML)

Delta-machine learning provides a cost-effective approach for developing high-level potential energy surfaces by leveraging the strengths of both low-level and high-level QM calculations [4]. In this framework, a ML model is trained to predict the difference (Δ) between a highly accurate but computationally expensive QM method and a less accurate but computationally inexpensive method [4].

The Δ-ML workflow involves:

• Generating a large dataset of molecular configurations with low-level QM calculations
• Computing high-level QM corrections for a strategically chosen subset
• Training a ML model to predict the difference between low-level and high-level results
• Applying the trained Δ model to new configurations, effectively upgrading low-level predictions to high-level accuracy

This approach was successfully applied to the H + CH4 hydrogen abstraction reaction, with resulting surfaces accurately reproducing both kinetics information from variational transition state theory and dynamics information from quasiclassical trajectory calculations [4].

Graph Neural Networks for Quantum Chemistry

Graph neural networks (GNNs) represent molecules as graphs with atoms as nodes and bonds as edges, enabling direct learning on molecular structures [7]. OrbNet, developed through a partnership between Caltech and Entos Inc., implements a particularly innovative GNN architecture that organizes electron orbitals as nodes and their interactions as edges [6]. This design has a natural connection to the Schrödinger equation and enables the model to perform accurately on molecules up to 10 times larger than those present in the training data [6].

Table: Comparison of Major ML Surrogate Approaches for Quantum Chemistry

Method	Key Innovation	Training Data Requirements	Accuracy Performance	Primary Applications
GAP-RSS Framework [2]	Combines Gaussian approximation potentials with random structure searching	~500-5,000 DFT single-point evaluations per system	~0.01 eV/atom for simple systems	Materials modeling, phase transitions, polymorph exploration
OrbNet [6]	Uses molecular orbitals as graph nodes rather than atoms	~100,000 reference QM calculations	Near-DFT accuracy for molecules 10x larger than training set	Molecular property prediction, reaction prediction, protein-ligand binding
Δ-ML [4]	Learns difference between high-level and low-level QM methods	Large low-level dataset + smaller high-level correction subset	Reproduces high-level kinetics and dynamics	Reaction barrier prediction, potential energy surface refinement
Molecular Set Representation [7]	Treats molecules as sets of atoms rather than connected graphs	Similar to GNNs	Matches or surpasses GNN performance on benchmark datasets	Drug discovery, materials science, bioactivity prediction

Automated Workflows for Potential Energy Surface Exploration

The Autoplex Framework

The development of high-quality MLIPs has traditionally been hampered by the manual generation and curation of training data [2]. The autoplex framework ("automatic potential-landscape explorer") addresses this bottleneck by automating the exploration and fitting of potential-energy surfaces [2]. Implemented as an openly available software package, autoplex integrates with existing computational materials science infrastructure and provides end-user-friendly workflows for high-throughput MLIP generation [2].

Autoplex employs iterative exploration through data-driven random structure searching (RSS), using gradually improved potential models to drive searches without relying on first-principles relaxations [2]. This approach requires only DFT single-point evaluations rather than full relaxations, significantly reducing computational overhead. The framework has demonstrated wide-ranging capabilities across diverse systems including the titanium-oxygen system, SiO2, crystalline and liquid water, and phase-change memory materials [2].

Workflow and Validation

The automated workflow for ML-surrogate development involves several interconnected stages, from initial data generation to final model validation, with iterative refinement based on active learning.

The performance of automatically generated MLIPs is rigorously validated against reference QM calculations. For example, in the titanium-oxygen system, autoplex achieved accuracies on the order of 0.01 eV/atom for relevant crystalline polymorphs with only a few thousand DFT single-point evaluations [2]. The framework's flexibility in handling varying stoichiometric compositions enables the development of unified potentials for entire chemical systems rather than individual compounds [2].

Table: Accuracy of Autoplex-Generated Potentials for Selected Systems

System	Target Structure	DFT Single-Point Evaluations	Final Energy Error (eV/atom)
Elemental Silicon [2]	Diamond-type structure	~500	~0.01
Elemental Silicon [2]	β-tin-type structure	~500	~0.01 (slightly higher)
Elemental Silicon [2]	oS24 allotrope	Few thousand	~0.01
TiOâ‚‚ Polymorphs [2]	Rutile, Anatase	Few thousand	~0.01
TiOâ‚‚ Polymorphs [2]	Bronze-type (B-)	Few thousand	Few tens of meV
Full Ti-O System [2]	Multiple stoichiometries	>5,000	~0.01

Molecular Representation Strategies

The performance of ML surrogates for QM calculations depends critically on how molecular structures are represented as input to the models. Different representation strategies emphasize different aspects of chemical structure, with significant implications for model accuracy, data efficiency, and transferability.

Graph-Based Representations

Graph-based representations treat molecules as graphs with atoms as nodes and bonds as edges, making them particularly suitable for GNN architectures [7]. This approach explicitly encodes molecular topology and has become widely adopted for molecular property prediction [7]. However, conventional graph representations face limitations in capturing complex bonding situations such as conjugated systems, ionic and metallic bonds, and dynamic intermolecular interactions [7].

Molecular Set Representation Learning

An emerging alternative treats molecules as sets (multisets) of atoms rather than connected graphs [7]. In this approach, each atom is represented as a vector of one-hot encoded atom invariants similar to those used in extended-connectivity fingerprints, with no explicit information about molecular topology [7]. This representation requires permutation-invariant neural network architectures such as DeepSets or Set-Transformers to handle variable-sized, unordered sets [7].

Comparative studies have shown that molecular set representation learning can match or surpass the performance of established graph-based methods across diverse domains including chemistry, biology, and materials science [7]. The performance of simple set-based models suggests that explicitly defined chemical bonds may not be as critical for many molecular learning tasks as previously assumed [7].

Orbital-Based Representations

OrbNet introduces a fundamentally different representation based on molecular orbital interactions rather than atomic connectivity [6]. By building a graph where nodes represent electron orbitals and edges represent interactions between orbitals, OrbNet establishes a more direct connection to the Schrödinger equation [6]. This domain-specific representation enables the model to extrapolate accurately to molecules much larger than those in its training set, overcoming a key limitation of standard deep learning models that typically only interpolate within their training data [6].

Essential Datasets and Benchmarking

The QM40 Dataset

The growing popularity of ML in molecular science has highlighted the scarcity of high-quality, chemically diverse datasets for training and benchmarking [8]. The QM40 dataset addresses this challenge by representing 88% of the FDA-approved drug chemical space, containing 162,954 molecules with 10 to 40 atoms composed of elements commonly found in drug structures (C, O, N, S, F, Cl) [8]. This represents a significant expansion over earlier datasets like QM9, which captures only 10% of drug-relevant chemical space due to its restriction to smaller molecules [8].

QM40 provides 16 key quantum mechanical parameters calculated at the B3LYP/6-31G(2df,p) level of theory, ensuring consistency with established datasets like QM9 and Alchemy [8]. Beyond standard QM properties, QM40 includes unique features such as local vibrational mode force constants as quantitative measures of bond strength, providing valuable resources for benchmarking both existing and new methods for predicting QM calculations using ML techniques [8].

Dataset Curation and Validation

The curation of QM40 followed a rigorous workflow to ensure data quality:

• Molecular SMILES strings from the ZINC database were converted to 3D structures using RDKit
• Initial geometries were pre-optimized using the extended tight-binding (xTB) method with GFN2-xTB level of theory
• DFT calculations were performed using Gaussian16 at the B3LYP/6-31G(2df,p) level
• Frequency calculations and local vibrational mode analysis were conducted using the LModeA software package
• Molecules with convergence failures, imaginary frequencies, or unphysical parameters were systematically excluded [8]

This meticulous validation process ensures that optimized geometries correspond to the original molecular structures and that all quantum chemical results are physically meaningful, providing a reliable foundation for training ML surrogates [8].

Table: Essential Software and Data Resources for ML-Surrogate Development

Resource	Type	Primary Function	Relevance to ML Surrogates
autoplex [2]	Software Framework	Automated exploration and fitting of potential-energy surfaces	High-throughput generation of MLIPs with minimal manual intervention
OrbNet [6]	Pre-trained Model / Architecture	Quantum chemistry calculations using symmetry-adapted atomic-orbital features	Accurate property prediction for molecules larger than training set
Gaussian16 [8]	Quantum Chemistry Software	Electronic structure calculations using DFT and other methods	Generation of reference data for training ML models
LModeA [8]	Analysis Tool	Local vibrational mode analysis and bond strength quantification	Provides unique features for dataset enhancement and model training
QM40 Dataset [8]	Benchmark Dataset	162,954 drug-like molecules with QM properties	Training and benchmarking for drug discovery applications
RDKit [8]	Cheminformatics Library	Molecular representation conversion and manipulation	Preprocessing of molecular structures for ML input

Machine learning surrogates have fundamentally transformed the landscape of computational chemistry and materials science by overcoming the traditional trade-off between computational cost and quantum-mechanical accuracy. Frameworks like autoplex demonstrate that the development of robust machine-learned interatomic potentials can be largely automated, making quantum-accurate atomistic modeling accessible to non-specialists [2]. Meanwhile, approaches like OrbNet and molecular set representation learning are expanding the boundaries of what ML surrogates can achieve, enabling accurate predictions for molecular systems significantly beyond their training data [6] [7].

As the field advances, several promising directions are emerging. The development of "foundational" MLIPs pre-trained on extensive datasets covering broad regions of chemical space represents a shift toward models that can be efficiently fine-tuned for specific applications [2]. The integration of active learning strategies with automated workflow systems will further reduce the human effort required to generate high-quality training data [2]. Additionally, the creation of larger, more chemically diverse benchmark datasets like QM40 will continue to drive improvements in model accuracy and generalizability [8]. These advances collectively promise to make quantum-mechanical accuracy routinely accessible for molecular systems of practical interest across drug discovery, materials design, and catalyst development.

The exploration of potential-energy surfaces (PES) is a fundamental challenge in computational materials science, physics, and chemistry, essential for understanding material properties and reaction mechanisms. Machine-learned interatomic potentials (MLIPs) have emerged as the preferred method for achieving quantum-mechanical accuracy in large-scale atomistic simulations. However, a significant bottleneck persists: the manual generation and curation of high-quality training data. This whitepaper introduces autoplex, an automated, open-source framework designed to overcome this bottleneck by enabling a hands-off, iterative workflow for exploring PES and fitting robust MLIPs. By leveraging data-driven random structure searching (RSS) and seamless integration with high-performance computing (HPC) systems, autoplex significantly accelerates the development of accurate, system-specific potentials, making high-fidelity atomistic modeling more accessible to the broader research community [2] [9] [10].

The Challenge of Potential-Energy Surface Exploration

A potential-energy surface represents the energy of a system as a function of its atomic coordinates. Navigating this hyper-surface to locate stable structures, transition states, and reaction pathways is critical for predicting material behavior. While foundational MLIPs trained on large datasets exist, they are not always suited for investigating specific, localized regions of chemical space or for studying systems with unique bonding environments. Building an MLIP from scratch for such tasks has traditionally required expert knowledge, manual configuration of training data, and labor-intensive active learning cycles, often relying on costly ab initio molecular dynamics [2] [9].

The autoplex framework directly addresses these challenges by automating the entire pipeline—from initial structure generation and quantum-mechanical evaluation to iterative model fitting and validation. This automation is a crucial step toward making ML-driven atomistic modelling a genuine mainstream tool [9] [10].

The autoplex Framework: Architecture and Core Components

The autoplex software is built as a modular set of tools that prioritizes interoperability with established computational materials science infrastructures. Its core architecture is designed for high-throughput operation on HPC systems [2].

Foundational Design Principles

• Interoperability: autoplex is designed around the core principles of the atomate2 workflow framework, which also underpins the Materials Project initiative. This ensures compatibility with a wide ecosystem of computational tools [2] [9] [10].
• Automation and High-Throughput: The framework automates the execution and monitoring of tens of thousands of individual computational tasks, a process that would be practically impossible to manage manually [2].
• Modularity and Extensibility: Although the current implementation prominently features the Gaussian Approximation Potential (GAP) framework due to its data efficiency, autoplex is architecturally designed to accommodate other MLIP fitting methodologies [2] [9].

The Core Workflow: From Random Search to Refined Potential

The following diagram illustrates the automated, iterative workflow at the heart of autoplex.

Diagram 1: The autoplex automated iterative workflow for MLIP development.

The workflow, depicted in Diagram 1, operates as a closed-loop system:

• Initialization: The process begins with the user defining the chemical system of interest [10].
• Random Structure Search (RSS): A diverse set of atomic configurations is generated automatically. This step is crucial for exploring both minima and "unfavourable regions" of the PES, which must be taught to the potential for robustness [2] [9].
• DFT Single-Point Evaluation: A limited number (e.g., 100 per iteration) of these configurations are selected for quantum-mechanical evaluation using Density-Functional Theory (DFT). A key innovation is that autoplex requires only single-point calculations, bypassing the need for computationally expensive DFT-based relaxations or pre-existing force fields [2] [9].
• MLIP Training/Update: The results from the DFT calculations are added to the training dataset, and a new MLIP (initially, a GAP model) is fitted or an existing one is updated [2].
• Validation and Convergence Check: The updated model's accuracy is validated against a target metric (e.g., a root mean square error (RMSE) of 0.01 eV/atom for energies). If the model has not converged, the improved potential is used to drive the next round of RSS, creating a self-improving cycle [2] [9].

Capability Demonstrations and Performance Metrics

The autoplex framework has been rigorously validated across a range of systems, from simple elements to complex binary compounds. The table below summarizes its performance in reproducing the energies of various crystalline phases.

Table 1: Performance of GAP-RSS Models Trained via autoplex [2] [9]

System	Compound	Structure Type	Final Energy RMSE (meV/atom)	Key Insight
Elemental	Silicon	Diamond-type	0.1	Highly symmetric structures learned rapidly (<500 evaluations) [9]
	Silicon	$\beta$-tin-type	~1.0	Higher-pressure phase with slightly higher error [9]
	Silicon	oS24	~10	Metastable, lower-symmetry phase requires more iterations [9]
Binary Oxide	TiO$_2$	Anatase	0.1 - 0.7	Common polymorphs are accurately captured [2] [9]
	TiO$_2$	Rutile	0.2 - 1.8	Common polymorphs are accurately captured [2] [9]
	TiO$_2$	TiO$_2$-B (Bronze)	20 - 24	More complex polymorph is harder to "learn" [2] [9]
Full Binary System	Ti$2$O$3$	Al$2$O$3$-type	9.1	Accurate description requires training on the full system [2] [9]
	TiO	Rocksalt (NaCl)	0.6	Accurate description requires training on the full system [2] [9]
	Ti$3$O$5$	Ti$3$O$5$-type	19	Model trained only on TiO$_2$ fails for this composition [2] [9]

Key Findings from Validation Studies

• Progressive Learning: The model's accuracy improves iteratively. For instance, the energy error for the oS24 silicon allotrope decreases systematically over several thousand DFT single-point evaluations [2] [9].
• Stoichiometric Flexibility: A critical demonstration involved the titanium-oxygen system. A model trained solely on TiO$2$ data failed catastrophically (errors >1 eV/atom) when applied to other stoichiometries like TiO or Ti$2$O. In contrast, a single autoplex workflow trained on the full Ti-O system delivered high accuracy across all these phases, highlighting the framework's power and flexibility [2] [9].
• Wide Applicability: The framework has also been successfully applied to other systems, including SiO$_2$, crystalline and liquid water, and phase-change memory materials, demonstrating its general utility [9] [10].

Table 2: Key Research Reagent Solutions for autoplex Workflows

Item	Function in Workflow	Key Details
autoplex Software	Core automation framework.	Open-source package available on GitHub. Provides high-throughput workflows for PES exploration and MLIP fitting [2] [10].
atomate2	Workflow management infrastructure.	Provides the underlying automation engine that autoplex leverages for job scheduling and task management [2] [10].
Gaussian Approximation Potential (GAP)	Primary MLIP engine.	A data-efficient kernel-based method for interatomic potentials. Used as the default fitting model within autoplex [2] [9].
Density-Functional Theory (DFT)	Source of quantum-mechanical reference data.	Used for single-point energy and force calculations. autoplex is agnostic to the specific DFT code used [2] [9].
Random Structure Searching (RSS)	Configurational space explorer.	Generates diverse atomic configurations for training. The GAP-RSS approach unifies searching with MLIP fitting [2] [9].

Experimental Protocol: A Step-by-Step Methodology

This section outlines a detailed protocol for running an autoplex workflow, using the titanium-oxygen system as a case study.

Pre-experiment Configuration and Setup

• Software Installation: Install the autoplex package from its public GitHub repository, ensuring all dependencies (atomate2, GAP, DFT code) are properly configured on the HPC environment [2] [10].
• System Definition: Define the chemical system in the input files. For a full binary exploration, specify the elements (Ti, O) and the desired stoichiometric ranges. No pre-existing training data is required.
• Computational Parameters: Set the DFT parameters (exchange-correlation functional, plane-wave cutoff, k-point mesh) and GAP fitting hyperparameters. These can be adopted from provided tutorials for consistency [9].

Workflow Execution and Data Acquisition

• Workflow Launch: Initiate the autoplex workflow with a few lines of code, as provided in the accompanying tutorials [10]. The workflow automatically submits jobs to the HPC scheduler.
• Iterative Cycle: The framework autonomously executes the loop shown in Diagram 1.
  • Step 1 - RSS Generation: Generates ~100+ new candidate structures per iteration.
  • Step 2 - DFT Single-Point: Selects structures for DFT calculation, extracting energies and forces.
  • Step 3 - MLIP Training: Adds new data to the training set and retrains the GAP model.
  • Step 4 - Validation: The model is tested against known phases (e.g., rutile, anatase) to compute the RMSE.
• Convergence Monitoring: The process continues iteratively until the target accuracy (e.g., an RMSE of 0.01 eV/atom for a selection of known phases) is achieved. For a complex binary system, this may require several thousand single-point evaluations [2] [9].

Output Analysis and Model Validation

• Final Model: The primary output is a finalized, robust GAP MLIP file ready for use in molecular dynamics or structure relaxation simulations.
• Performance Validation: The model's performance is quantified using tables like Table 1, comparing predicted versus DFT-calculated energies for a suite of benchmark structures not included in the training set.
• Application: The validated potential can be deployed for large-scale simulations to explore finite-temperature properties, phase transitions, or chemical reactions with near-DFT accuracy.

The autoplex framework represents a significant advancement in the automation of machine learning for atomistic simulations. By integrating random structure searching, on-the-fly quantum-mechanical evaluation, and iterative model fitting into a single, streamlined workflow, it effectively addresses the critical bottleneck of data generation in MLIP development. As demonstrated by its successful application to a diverse set of materials, autoplex provides researchers with a powerful, hands-off tool for building accurate and robust interatomic potentials from scratch. This automation not only accelerates research but also lowers the barrier to entry, paving the way for the broader adoption of high-fidelity machine learning potentials across physics, chemistry, and materials science [2] [9] [10].

The exploration of potential energy surfaces (PES) is fundamental to predicting material properties and biological interactions. Machine learning (ML) has emerged as a transformative tool for this task, enabling high-accuracy simulations at a fraction of the computational cost of traditional quantum mechanical methods. Machine-learned interatomic potentials (MLIPs) map atomic configurations to their energies and forces, creating surrogates that approximate quantum-mechanical accuracy for large-scale systems [2]. This technical guide examines core applications of this approach across two domains: the identification of TiO2 polymorphs and the prediction of biomolecular system transformations.

The automation of MLIP development is accelerating this field. Frameworks like autoplex automate the exploration and fitting of PES, using iterative random structure searching (RSS) and active learning to build robust models with minimal human intervention [2]. Similarly, the ænet package provides open-source tools for constructing artificial neural network (ANN) potentials, as demonstrated for bulk TiO2 [11]. These tools are pushing the boundaries of what is computationally feasible in materials and biomolecular modeling.

Core Applications and Quantitative Performance

The following case studies demonstrate the performance of machine learning in predicting material and biological properties.

Table 1: Performance of ML Models in Material and Biomolecular Applications

Application Domain	ML Model	Key Performance Metrics	Reference / System
TiO2 Polymorph Identification	CNN-LSTM Hybrid	Top-1 Accuracy: 99.12%; Top-5 Accuracy: 99.30%	RRUFF Dataset [12]
Photocatalytic Degradation Prediction	XGBoost (XGB)	R² (test): 0.936; RMSE (test): 0.450 min⁻¹/cm²	Air Contaminants [13]
	Decision Tree (DT)	R² (test): 0.924; RMSE (test): 0.494 min⁻¹/cm²	Air Contaminants [13]
	Lasso Regression (LR2)	R² (test): 0.924; RMSE (test): 0.490 min⁻¹/cm²	Air Contaminants [13]
Pathology Prediction (TiO2 NPs)	Supervised ML with SMOTE	Accuracy: 0.89; Precision: 0.90; Recall: 0.88	17-Gene Biomarker Panel [14]
PES Exploration (TiO2)	Gaussian Approximation Potential (GAP)	Energy RMSE: ~0.01 eV/atom for rutile, anatase [2]	Titanium-Oxygen System [2]

Detailed Methodologies and Experimental Protocols

Deep-Learning for Raman Spectroscopy-Based Polymorph Identification

This protocol details the automated identification of TiO2 polymorphs from Raman spectra using a hybrid deep-learning model, eliminating the need for expert-guided pre-processing [12].

• Step 1: Data Acquisition and Preparation. The model is trained and evaluated using Raman spectra from the publicly available RRUFF spectral database. For experimental validation, TiO2 polymorphs such as Anatase, Rutile, and P25 Degussa are synthesized or sourced.
• Step 2: Model Architecture and Training. The framework uses a combination of 1D Convolutional Neural Networks (CNNs) and Long Short-Term Memory (LSTM) networks.
  • The input is a one-dimensional Raman spectrum.
  • Four 1D Convolutional layers with a kernel size of 2 and ReLU activation extract local feature patterns.
  • Convolutional layers are followed by max-pooling layers (pool size of 2) for dimensionality reduction.
  • The feature sequence is then processed by an LSTM layer to model long-range dependencies and temporal patterns in the spectrum.
  • The output of the LSTM is flattened and passed through fully-connected dense layers with ReLU activation.
  • The final output layer uses a SoftMax activation to assign probabilities to each polymorph class.
• Step 3: Validation. The trained model is validated by predicting the identity of synthesized TiO2 samples and comparing the results to known ground truths, achieving high-confidence identification even for defect-rich Anatase and modified Rutile [12].

Automated Exploration of Potential-Energy Surfaces withautoplex

The autoplex framework automates the development of machine-learned interatomic potentials (MLIPs) for exploring complex material systems like Ti–O [2].

• Step 1: Initialization and Random Structure Searching (RSS). The process begins by generating a set of random initial atomic configurations for the chemical system of interest (e.g., Ti–O). No pre-existing potential or extensive user-provided structures are required.
• Step 2: Active Learning and MLIP Training.
  • Configuration Sampling: Molecular Dynamics (MD) simulations are run using a current version of the MLIP (e.g., a Gaussian Approximation Potential, GAP). Structures that venture into unexplored regions of the configuration space are saved.
  • DFT Single-Point Calculations: The energies, forces, and stresses of these new configurations are computed using Density-Functional Theory (DFT) to create high-quality reference data.
  • Model Retraining: The MLIP is retrained on the aggregated and curated dataset. This iterative loop of sampling, DFT calculation, and retraining continues until no new configurations are found for a set number of iterations, indicating convergence.
• Step 3: Validation and Application. The final, robust MLIP can be used to accurately predict the stability and properties of known and newly discovered phases across the chemical system, as demonstrated for TiO2 polymorphs and sub-oxides like Ti2O3 and TiO [2].

Predicting TiO2 Nanoparticle-Induced Pathology from Transcriptomics

This methodology applies supervised machine learning to predict pulmonary pathology from gene expression changes induced by TiO2 nanoparticles (NPs) [14].

• Step 1: Data Collection and Preprocessing. A dataset is constructed from transcriptomic analyses of lung tissue from mice exposed to various rutile-type TiO2 NPs. The data includes NP characteristics (primary size, surface area, surface charge) and post-exposure duration. A set of 621 differentially expressed genes is identified.
• Step 2: Model Training with Imbalance Mitigation.
  • The genes are classified as responsive or non-responsive to NP exposure.
  • To address class imbalance, the Synthetic Minority Oversampling Technique (SMOTE) is applied, generating synthetic data points for the underrepresented classes.
  • A battery of supervised ML models (e.g., SVM, Random Forest, etc.) is trained on this balanced dataset to predict gene expression changes based on NP properties.
• Step 3: Biomarker Identification and Model Validation. The most accurate models are selected based on metrics like accuracy, precision, and recall. These models identify a core set of 17 transcriptomic biomarkers (e.g., Saa3, Ccl2, IL-1β). The models are subsequently validated on an independent test dataset to ensure predictive reliability for lung inflammation and fibrosis pathways [14].

Table 2: Key Computational Tools and Databases for ML-driven PES Exploration

Resource Name	Type	Primary Function	Application Example
RRUFF Database	Spectral Database	Repository of reference Raman spectra for mineral identification.	Training data for CNN-LSTM model for TiO2 polymorphs [12].
autoplex Software	MLIP Workflow Tool	Automated framework for exploring and fitting potential-energy surfaces.	Building GAP models for the Ti-O system [2].
ænet Package	Software Package	Open-source tool for constructing and using Artificial Neural Network (ANN) potentials.	Creating an ANN potential for bulk TiO2 structures [11].
Gaussian Approximation Potential (GAP)	MLIP Method	A data-efficient MLIP framework based on Gaussian process regression.	Driving RSS and potential fitting in autoplex [2].
Moment Tensor Potentials (MTP)	MLIP Method	A MLIP implementation using moment tensors to describe atomic environments.	Predicting stable 2D Mo-S structures on a substrate [15].
SMOTE	Data Preprocessing Algorithm	Synthesizes new instances of minority classes to correct dataset imbalance.	Improving prediction of active vs. non-active gene responses to TiO2 NPs [14].

The integration of machine learning into the exploration of potential energy surfaces provides a unified and powerful framework for advancing both materials science and biomolecular research. The techniques detailed here—from deep learning for spectral analysis to automated MLIP development—enable the rapid, accurate, and insightful prediction of properties and behaviors in complex systems like TiO2 polymorphs and biomolecular coronas. As computational tools and automated frameworks continue to mature and become more accessible, they will undoubtedly become a standard component in the toolkit of researchers and industrial scientists, accelerating the discovery and design of new materials and therapeutic agents.

The Critical Need for High-Quality, Diverse Training Data

Machine-learned potential energy surfaces (ML-PESs) have emerged as a transformative tool, enabling large-scale atomistic simulations with quantum-mechanical accuracy across diverse fields, from high-pressure research and molecular reaction mechanisms to the realistic modelling of proteins [2]. The fundamental promise of ML-PESs is to overcome the long-standing accuracy-versus-efficiency trade-off that hampers traditional approaches in computational materials science and chemistry [16]. However, the performance, reliability, and ultimate success of these machine learning (ML) models are not guaranteed by the sophistication of the algorithm alone. They hinge critically on a more foundational element: the quality, quantity, and diversity of the training data. The process of generating and curating this data has historically been a major bottleneck, often requiring manual, time-intensive efforts and deep domain expertise [2]. This whitepaper examines the central role of training data in the exploration of potential energy surfaces, detailing the challenges, methodologies, and practical protocols for constructing robust datasets that yield accurate, generalizable, and physically meaningful ML models.

The Data Bottleneck in ML-PES Development

The development of ML-PESs is a multi-step procedure where data-related challenges permeate every stage [17]. Traditionally, these potentials were hand-crafted models built from configurations manually tailored for specific domain tasks [2]. This process is not only slow but also susceptible to human bias, which can lead to datasets that lack the diversity required for the model to explore the full configurational space of the system.

A primary challenge is the source of inaccuracy. Most ML-PESs are trained on data generated from Density Functional Theory (DFT) calculations, which are more affordable but less accurate than higher-level methods like CCSD(T). Consequently, the ML potential inherits these inaccuracies and may not achieve quantitative agreement with experimental observations [16]. For instance, a previous ML model for titanium failed to quantitatively reproduce experimental temperature-dependent lattice parameters and elastic constants, with deviations attributed directly to the underlying DFT functional [16].

Furthermore, the scale and scope of the data present another significant hurdle. Generating ab initio data that is simultaneously accurate, large in volume, and broad in scope (to avoid distribution shift) is exceptionally challenging [16]. Due to the computational cost of DFT, simulations are typically limited to system sizes of a few hundred atoms, raising questions about whether long-range interactions can be adequately learned from such constrained data [16]. An ML-PES trained on a narrow set of configurations, such as only one stoichiometry in a binary system, will inevitably fail when applied to other phases or compositions, leading to unacceptably high errors [2].

Strategies for Automated and Diverse Data Generation

To overcome the limitations of manual data curation, automated and data-driven strategies are essential for the efficient exploration of complex potential-energy landscapes.

Automated Random Structure Searching

The autoplex framework exemplifies the trend toward automation. It implements an automated approach to iterative exploration and MLIP fitting through data-driven random structure searching (RSS) [2]. Its design philosophy emphasizes interoperability with existing software architectures and ease of use for the end-user. The core principle involves using gradually improved machine-learned potentials to drive random structure searches, requiring only DFT single-point evaluations rather than costly ab initio molecular dynamics relaxations [2]. This method has demonstrated wide-ranging capabilities, successfully exploring systems from elemental silicon and polymorphs of TiOâ‚‚ to the full binary titanium-oxygen system [2].

Fused Data Learning

An orthogonal and powerful strategy is fused data learning, which leverages both simulation data and experimental measurements to train a single ML potential. This approach concurrently uses a DFT trainer, which performs standard regression on quantum-mechanical data, and an EXP trainer, which optimizes model parameters to match experimental observables using methods like Differentiable Trajectory Reweighting (DiffTRe) [16]. This methodology corrects for known inaccuracies of DFT functionals against target experimental properties, resulting in a molecular model of higher overall accuracy compared to models trained on a single data source [16].

Table 1: Performance Comparison of ML-PES Training Strategies for Titanium

Training Strategy	Description	Key Outcome
DFT Pre-trained	Trained only on DFT-calculated energies, forces, and virial stress [16].	Achieves chemical accuracy on DFT test data but may disagree with key experimental properties [16].
DFT & EXP Fused	Concurrent training on both DFT data and experimental properties (e.g., elastic constants) [16].	Satisfies all target objectives (DFT and experiment); results in a model of higher, more consistent accuracy [16].

Experimental Protocols and Workflows

The autoplex Workflow for Automated Data Generation

The autoplex framework provides a concrete protocol for automated data generation and potential fitting. The following diagram illustrates its iterative, closed-loop workflow:

Diagram 1: Automated Exploration and Learning Workflow

Protocol Steps:

• Initialization: Begin with an initial, potentially small, dataset of atomic configurations and their corresponding ab initio energies and forces.
• Random Structure Search (RSS): Generate a diverse set of new atomic configurations through random structure searching, driven by the current best ML potential [2].
• DFT Single-Point Calculations: Perform high-throughput DFT single-point evaluations on the newly proposed structures to obtain quantum-mechanical reference data. This avoids the high cost of ab initio molecular dynamics relaxations [2].
• Dataset Curation & Expansion: Add the new data points to the growing training dataset.
• ML Model Training: Train or refine the machine-learned interatomic potential (e.g., a Gaussian Approximation Potential) on the expanded dataset [2].
• Validation and Iteration: Evaluate the model's accuracy on a set of known reference structures or properties. If the accuracy target (e.g., ~0.01 eV/atom) is not met, the loop continues from step 2 [2].

The Fused Data Learning Protocol

For systems where agreement with experimental data is critical, the fused data learning protocol is more appropriate. The workflow for this strategy is depicted below:

Diagram 2: Fused Data Training Workflow

Protocol Steps:

• Pre-training: Initialize the ML potential by training it on a comprehensive database of DFT calculations (energies, forces, virial stresses) [16]. This provides a physically reasonable starting point.
• DFT Trainer Epoch: Perform one epoch of training using the DFT database. The loss function minimizes the difference between the ML potential's predictions and the DFT-calculated energies, forces, and stresses [16].
• EXP Trainer Epoch: Perform one epoch of training using experimental data.
  • Run molecular dynamics simulations using the current ML potential to compute macroscopic properties (e.g., elastic constants, lattice parameters).
  • Use the DiffTRe method to calculate the gradients of the difference between simulated and experimental properties with respect to the ML potential's parameters.
  • Update the model parameters to reduce this difference [16].
• Iteration: Alternate between the DFT and EXP trainers until the model converges and satisfactorily reproduces both the quantum-mechanical and experimental target properties [16].

Quantitative Benchmarks and Performance

The effectiveness of these advanced data generation strategies is demonstrated by their performance on real chemical systems. The iterative approach of autoplex shows a clear learning curve for increasingly complex structures.

Table 2: autoplex Performance on Test Systems (Target Accuracy: <0.01 eV/atom) [2]

System	Example Structure	Structures to Target Accuracy	Notes
Elemental Silicon	Diamond-type	~500	Highly symmetric structures learned rapidly [2].
	β-tin-type	~500	Slightly higher error than diamond-type [2].
	oS24 allotrope	Few thousand	Metastable, lower-symmetry phase requires more data [2].
TiOâ‚‚ Polymorphs	Rutile & Anatase	~1,000	Common polymorphs learned efficiently [2].
	TiOâ‚‚-B (bronze)	>1,000	More complex connectivity requires greater sampling [2].
Ti–O System	Ti₂O₃, TiO, Ti₂O	Several thousand	Complex stoichiometries and electronic structures demand extensive exploration [2].

Furthermore, the fused data approach provides a direct path to correcting systematic errors. Research on a titanium potential showed that a model trained only on DFT data (DFT pre-trained) could not accurately reproduce experimental elastic constants across a range of temperatures. In contrast, the DFT & EXP fused model successfully matched these experimental targets while maintaining low errors on the DFT test dataset, proving that the model was not merely "forgetting" the quantum-mechanical data [16].

The Scientist's Toolkit: Essential Research Reagents

Building a high-quality ML-PES requires a suite of software tools and data resources. The following table details key "research reagents" essential for work in this field.

Table 3: Essential Tools for ML-PES Development

Tool / Resource	Type	Primary Function	Reference
autoplex	Software Framework	Automated workflow for exploring and fitting potential-energy surfaces via random structure searching [2].	[2]
Gaussian Approximation Potential (GAP)	ML Potential Framework	A kernel-based potential used for its data efficiency in driving exploration and potential fitting [2].	[2]
DiffTRe	Algorithm/Method	Enables top-down training of ML potentials on experimental data without backpropagation through entire simulations [16].	[16]
Graph Neural Network (GNN) Potentials	ML Model Architecture	A class of high-capacity neural network potentials (e.g., used in fused data learning) suitable for complex materials [16].	[16]
Materials Project Database	Data Resource	A source of diverse crystalline structures and properties, often used for training "foundational" ML potentials [2].	[2]
Active Learning Scripts	Software Component	Algorithms for on-the-fly selection of new configurations for DFT evaluation to expand the training dataset optimally [16].	[16]
Tofisopam	Tofisopam CAS 22345-47-7 - For Research Use		Bench Chemicals
URB-597	URB-597, CAS:546141-08-6, MF:C20H22N2O3, MW:338.4 g/mol	Chemical Reagent	Bench Chemicals

The exploration of potential energy surfaces with machine learning has reached a stage where the model architecture is no longer the primary limiting factor. The critical determinant of success is the quality and diversity of the training data. As evidenced by the development of automated frameworks like autoplex and innovative training strategies like fused data learning, the field is moving decisively to address the data bottleneck. These approaches systematically generate broad and relevant datasets, incorporate physical validity through experimental data, and minimize human bias through automation. For researchers and drug development professionals, adopting these methodologies is paramount. The construction of robust, reliable, and transferable ML-PESs depends on a foundational commitment to building superior training datasets, which in turn enables more confident discovery and design of new molecules and materials.

Building Robust ML-PES: Architectures, Strategies, and Real-World Applications in Biomedicine

In the pursuit of exploring complex potential-energy surfaces (PES) for computational materials science and drug discovery, researchers are faced with a critical choice: employing sophisticated deep neural networks (DNNs) or leveraging robust kernel-based methods. This decision significantly impacts not only the predictive accuracy but also the computational efficiency, data requirements, and interpretability of the resulting models. Machine learning has become ubiquitous in materials modelling, enabling large-scale atomistic simulations with quantum-mechanical accuracy [2]. However, developing these machine-learned interatomic potentials requires high-quality training data, and the manual generation and curation of such data can be a major bottleneck [2].

The field is currently witnessing a trend toward automation and hybridization. Automated frameworks like autoplex ('automatic potential-landscape explorer') are emerging to streamline the exploration and fitting of potential-energy surfaces [2] [18]. Simultaneously, hybrid approaches such as Δ-machine learning (Δ-ML) are demonstrating remarkable cost-effectiveness for developing high-level potential energy surfaces from low-level configurations [4]. This guide examines the fundamental characteristics, relative strengths, and optimal application domains for both kernel-based and neural network approaches within the specific context of PES exploration and drug discovery applications.

Theoretical Foundations: How Kernel Methods and Neural Networks Work

Kernel Methods: The Power of Feature Space Transformation

Kernel methods, such as Kernel Ridge Regression (KRR) and Support Vector Machines (SVM), operate on a simple but powerful principle: they transform input data into a higher-dimensional feature space where complex nonlinear relationships become linearly separable. This transformation is performed implicitly through a kernel function, which computes the dot product between vectors in this new space without explicitly constructing the feature vectors themselves [19] [20].

The mathematical foundation lies in the kernel trick, which allows algorithms to express their computations in terms of inner products between all pairs of data points. For a kernel function (k(\mathbf{x}i, \mathbf{x}j)) and a set of training data ({(\mathbf{x}i, yi)}{i=1}^N), the prediction for a new point (\mathbf{x}) takes the form: [ f(\mathbf{x}_) = \sum{i=1}^N \alphai k(\mathbf{x}i, \mathbf{x}*) ] where (\alpha_i) are parameters learned from the data [20]. This formulation enables kernel methods to model complex relationships while remaining convex optimization problems with guaranteed global optima.

Neural Networks: Hierarchical Feature Learning

Neural networks, particularly deep architectures, learn hierarchical representations of data through multiple layers of nonlinear transformations. Each layer applies an affine transformation followed by a nonlinear activation function, allowing the network to progressively learn more abstract features from the raw input [21] [20].

A basic feedforward neural network with (L) layers transforms input (\mathbf{x}) as: [ \mathbf{h}^{(1)} = \phi(\mathbf{W}^{(1)}\mathbf{x} + \mathbf{b}^{(1)}) ] [ \mathbf{h}^{(l)} = \phi(\mathbf{W}^{(l)}\mathbf{h}^{(l-1)} + \mathbf{b}^{(l)}) \quad \text{for } l = 2, \ldots, L ] [ f(\mathbf{x}) = \mathbf{W}^{(L+1)}\mathbf{h}^{(L)} + \mathbf{b}^{(L+1)} ] where (\mathbf{W}^{(l)}) and (\mathbf{b}^{(l)}) are the weights and biases of layer (l), and (\phi) is a nonlinear activation function [21]. Unlike kernel methods with fixed transformations, neural networks learn the feature representation directly from data through backpropagation and gradient-based optimization.

Table: Core Architectural Differences Between Kernel Methods and Neural Networks

Aspect	Kernel Methods	Neural Networks
Feature Learning	Fixed, explicit transformation via kernel function	Learned, hierarchical representation through multiple layers
Optimization Landscape	Typically convex with global optimum guarantee	Non-convex with multiple local minima
Parameter Growth	Grows with training data size (N parameters)	Fixed architecture size (independent of data points)
Theoretical Basis	Statistical learning theory, Reproducing Kernel Hilbert Space	Universal approximation theorems, composition of functions
Implementation	Requires storing kernel matrix (O(N²) memory)	Forward/backward propagation through computational graph

Comparative Analysis: Performance, Data Efficiency, and Computational Requirements

Performance and Data Efficiency

Empirical evidence from scientific computing reveals that the relative performance of kernel methods versus neural networks is highly context-dependent. In neuroimaging applications, kernel regression has demonstrated competitive performance with DNNs for predicting individual phenotypes from whole-brain resting-state functional connectivity patterns, even across large sample sizes of nearly 10,000 participants [19]. This study revealed that kernel regression and three different DNN architectures achieved similar performance across a wide range of behavioral and demographic measures, with kernel regression incurring significantly lower computational costs [19].

For high-stationarity data, such as vehicle flow through tollbooths, classical machine learning algorithms like XGBoost can outperform more complex RNN-LSTM models, particularly in terms of MAE and MSE metrics [22]. This highlights how shallower algorithms sometimes achieve better adaptation to certain time series than much deeper models that tend to develop smoother predictions [22].

However, in materials science applications involving complex potential-energy surfaces, neural networks have demonstrated remarkable capabilities. The automated autoplex framework successfully uses machine-learned interatomic potentials (including neural network architectures) to explore configurational space for systems like titanium-oxygen, SiOâ‚‚, and phase-change memory materials [2]. For these applications, the data efficiency and accuracy of Gaussian approximation potentials (a kernel-based method) has proven particularly valuable for driving exploration and potential fitting [2].

Computational and Resource Requirements

The computational demands of these approaches differ significantly, influencing their practical applicability:

Table: Computational Requirements and Scaling Characteristics

Resource Factor	Kernel Methods	Neural Networks
Training Time	O(N³) time for matrix inversion, but often faster convergence	Can take days to weeks depending on complexity and architecture [21]
Inference Speed	O(N) per prediction (depends on support vectors)	O(1) after training (fixed computational graph)
Memory Usage	O(N²) for kernel matrix storage	O(W) for model parameters (W = number of weights)
Hardware Needs	Standard CPUs often sufficient	High-performance GPUs/TPUs typically required [21]
Data Scalability	Becomes prohibitive for >10⁵ samples	Scales to millions of data points with mini-batch training

Kernel methods face significant memory constraints for large datasets due to the kernel matrix growing quadratically with the number of training points. Neural networks, while more computationally intensive to train, offer constant-time prediction after training and can handle massive datasets through mini-batch optimization [21] [19].

Methodological Protocols: Implementation Guidelines

Kernel Method Implementation Protocol

Data Preprocessing and Kernel Selection:

• Standardize features to zero mean and unit variance, as kernel methods are sensitive to feature scales.
• Select an appropriate kernel function based on data characteristics:
  • Linear kernel: (k(\mathbf{x}, \mathbf{x}') = \mathbf{x}^\top\mathbf{x}')
  • Polynomial kernel: (k(\mathbf{x}, \mathbf{x}') = (\gamma\mathbf{x}^\top\mathbf{x}' + r)^d)
  • Radial Basis Function (RBF): (k(\mathbf{x}, \mathbf{x}') = \exp(-\gamma\|\mathbf{x} - \mathbf{x}'\|^2))
• For PES applications, consider designing problem-specific kernels that incorporate physical invariants or molecular symmetries.

Training and Validation:

• Solve the dual optimization problem for kernel ridge regression: [ \boldsymbol{\alpha} = (K + \lambda I)^{-1}\mathbf{y} ] where (K) is the kernel matrix and (\lambda) is the regularization parameter.
• Use cross-validation to optimize hyperparameters (kernel parameters, regularization strength).
• For large datasets, employ approximation techniques (Nyström method, random Fourier features) to reduce computational burden.

Neural Network Implementation Protocol

Architecture Design and Training:

• Select network architecture based on data modality:
  • Fully-connected networks for feature vectors [19]
  • Graph neural networks for molecular structures [2]
  • Convolutional networks for spatial data [21]
• Implement appropriate physical constraints:
  • Incorporate translational, rotational, and permutational invariances
  • Use physically motivated activation functions or output layers
  • Enforce conservation laws through architectural choices or loss functions
• Employ robust training procedures:
  • Use batch normalization for stable training
  • Implement learning rate scheduling and early stopping
  • Apply regularization techniques (dropout, weight decay) to prevent overfitting

The autoplex framework demonstrates a complete workflow for neural network potential training, combining automated structure searching with iterative model refinement [2]. This approach gradually improves potential models to drive searches without relying on first-principles relaxations at each iteration, requiring only DFT single-point evaluations [2].

Hybrid and Advanced Approaches

Δ-Machine Learning Protocol: Δ-machine learning (Δ-ML) represents a powerful hybrid approach that combines the benefits of computational efficiency and high accuracy [4]. The implementation protocol involves:

• Develop a low-level analytical potential that captures the basic physics of the system.
• Sample configurations using the low-level potential to generate training data.
• Train a machine learning model (neural network or kernel method) to learn the difference (Δ) between the low-level potential and high-level reference calculations.
• Combine predictions for final inference: [ E{\text{total}} = E{\text{low-level}} + \Delta_{\text{ML}} ]
• Validate performance on kinetics and dynamics properties, as demonstrated for the H + CHâ‚„ hydrogen abstraction reaction [4].

Neural Kernel Methods: Recent advances have introduced neural kernel methods that leverage the robustness and interpretability of kernel methods while generating data-dependent kernels tailored to specific needs [20]. The implementation involves:

• Train a neural network on the available data.
• Extract features from the penultimate layer of the network.
• Compute the neural kernel as the inner product between these feature representations.
• Perform kernel ridge regression using the neural kernel.
• This approach is particularly valuable for capturing high-dimensional constitutive responses of materials with complex internal structures [20].

Application to Potential Energy Surfaces and Drug Discovery

Potential Energy Surface Exploration

The exploration of potential energy surfaces represents a prime application area where the choice between kernel methods and neural networks has significant implications. The autoplex framework demonstrates how automated machine learning can accelerate PES exploration for systems ranging from elemental silicon to binary titanium-oxygen systems [2].

For silicon allotropes, including the diamond-type structure and higher-pressure forms like β-tin, machine-learned potentials can achieve accuracies on the order of 0.01 eV/atom with a few hundred DFT single-point evaluations [2]. More complex polymorphs, such as the open-framework oS24 allotrope, require a few thousand evaluations but remain tractable [2].

In the titanium-oxygen system, different stoichiometric compositions (Ti₂O₃, TiO, Ti₂O) present varying learning challenges. While simpler phases like rutile and anatase TiO₂ are learned quickly, achieving target accuracy for the full binary system requires more iterations as the search space increases in complexity [2]. This highlights the importance of selecting models that can handle the specific complexity of the target PES.

Table: Performance on Material Systems (Adapted from autoplex Framework [2])

Material System	Target Accuracy (eV/atom)	Structures Required	Recommended Approach
Elemental Silicon	0.01	~500	Gaussian Approximation Potentials
TiOâ‚‚ Polymorphs	0.01	~1000-2000	Neural Network Potentials
Binary Ti-O System	0.01	>5000	Hybrid or Iterative NN
Phase-Change Materials	0.01-0.05	Varies by complexity	Task-Specific Optimization

Drug Discovery Applications

In pharmaceutical research, both kernel methods and neural networks play crucial roles in accelerating drug discovery pipelines. AI-driven platforms have demonstrated remarkable efficiency, with companies like Exscientia reporting design cycles approximately 70% faster and requiring 10× fewer synthesized compounds than industry norms [23].

Target Identification and Validation:

• Kernel methods excel in early-stage target prediction using structured biological data
• Graph neural networks effectively model molecular interactions and protein-ligand binding
• Hybrid approaches combine strengths for multi-task learning across biological domains

Lead Optimization: This application stage dominates the machine learning in drug discovery market, holding approximately 30% share in 2024 [24]. Neural networks, particularly deep learning architectures, enable:

• Prediction of drug-target interactions and binding affinities
• Generative design of novel molecular structures with desired properties
• Optimization of ADMET (Absorption, Distribution, Metabolism, Excretion, Toxicity) profiles

Clinical Trial Design: The clinical trial design and recruitment segment is experiencing rapid growth in ML adoption [24]. Kernel methods support:

• Patient stratification using electronic health records
• Predictive modeling of clinical outcomes
• Optimization of trial protocols and recruitment strategies

Essential Research Reagents and Computational Tools

Successful implementation of kernel-based or neural network approaches for PES exploration requires specific computational tools and frameworks. The following table summarizes key resources mentioned in the research literature:

Table: Essential Research Tools for PES and Molecular Modeling

Tool/Framework	Type	Primary Function	Application Example
autoplex [2]	Software Package	Automated exploration and fitting of PES	Titanium-oxygen system, SiOâ‚‚, water
Gaussian Approximation Potential (GAP) [2]	Kernel Method	Machine-learned interatomic potentials	Iterative training with single-point DFT
Δ-ML Framework [4]	Hybrid Method	High-level PES from low-level configurations	H + CH₄ hydrogen abstraction reaction
Neural Kernel Method [20]	Hybrid Method	High-dimensional yield surface reconstruction	Micromorphic plasticity of layered materials
Atomate2 [2]	Workflow System	Computational materials science workflows	Integration with Materials Project data

The choice between kernel-based and neural network approaches for exploring potential-energy surfaces involves careful consideration of multiple factors, including data availability, computational resources, accuracy requirements, and interpretability needs. Kernel methods generally offer advantages for smaller datasets, provide stronger theoretical guarantees, and have lower computational requirements during training. Neural networks excel at handling very large, complex datasets and can automatically learn relevant features without extensive manual engineering.

Future developments in this field are likely to focus on several key areas:

• Increased automation in dataset construction and model training, as exemplified by the autoplex framework [2]
• Growth of hybrid models that combine traditional ML approaches with neural networks for improved performance [21]
• Development of more efficient neural architectures that require less computational power [21]
• Advancements in explainable AI to make neural network predictions more interpretable for scientific applications [21]
• Expansion of transfer learning and foundation models for materials science and drug discovery [23]

For researchers exploring potential energy surfaces, we recommend starting with kernel methods for systems with limited data or when interpretability is crucial, then progressing to neural networks as dataset size and complexity increase. Hybrid approaches like Δ-machine learning and neural kernels offer promising middle grounds that leverage the strengths of both paradigms. As the field continues to evolve, the integration of physical constraints and domain knowledge into both kernel-based and neural network models will be essential for advancing the exploration of complex potential-energy surfaces across materials science and drug discovery.

The exploration of potential energy surfaces (PES) stands as a fundamental challenge in computational materials science and drug discovery. Traditional quantum mechanical methods, while accurate, remain computationally prohibitive for the extensive sampling required for thorough PES exploration. Machine learning interatomic potentials (MLIPs) have emerged as transformative surrogates, bridging the accuracy of quantum mechanics with the efficiency of classical force fields [25]. Among these, universal MLIPs (uMLIPs) represent a paradigm shift—foundational models trained on massive datasets capable of handling diverse chemistries and structures without system-specific retraining [26] [27]. The integration of geometric equivariance, particularly through architectures like MACE and other equivariant graph neural networks (GNNs), has been instrumental in achieving this universality while maintaining physical consistency [28] [25]. This technical guide examines the architectural innovations, performance benchmarks, and methodological frameworks that enable these advanced models to accelerate the exploration of potential energy surfaces with unprecedented fidelity and efficiency.

Architectural Foundations: From Invariant Descriptors to Equivariant Representations

The Evolution of Geometric GNNs

Early MLIPs relied on handcrafted invariant descriptors—initially bond lengths, later incorporating bond angles and dihedral angles—to encode the potential energy surface [25]. While invariant to rotations and translations, these representations often struggled to distinguish structures with identical bond lengths but different overall configurations, or identical angles but different spatial arrangements [28]. The advent of equivariant architectures fundamentally addressed these limitations by explicitly embedding physical symmetries into the network structure itself.

Equivariant models explicitly maintain internal feature representations that transform predictably under rotations and translations according to the underlying symmetry group, guaranteeing that scalar predictions (e.g., total energy) remain invariant while vector and tensor targets (e.g., forces, dipole moments) exhibit correct equivariant behavior [25]. This approach parallels classical multipole theory in physics, encoding atomic properties as monopole, dipole, and quadrupole tensors and modeling their interactions through tensor products [25].

Efficient Equivariant Architectures

Modern equivariant architectures balance expressiveness with computational efficiency. The Efficient Equivariant Graph Neural Network (E2GNN) exemplifies this trend, employing a scalar-vector dual representation rather than computationally expensive higher-order tensor representations [28]. In E2GNN, each node maintains both scalar features ( \mathbf{x}i \in \mathbb{R}^F ) (invariant) and vector features ( \overrightarrow{\mathbf{x}}i \in \mathbb{R}^{F \times 3} ) (equivariant), updated through four key processes: global message distributing, local message passing, local message updating, and global message aggregating [28].

The local message passing in E2GNN combines information from neighboring nodes through symmetry-preserving operations:

[ \begin{align} \mathbf{m}_i &= \sum_{v_j \in \mathcal{N}(v_i)} (\mathbf{W}_h \mathbf{x}_j^{(t)}) \circ \lambda_h(\|\overrightarrow{\mathbf{r}}_{ji}\|) \ \overrightarrow{\mathbf{m}}_i &= \sum_{v_j \in \mathcal{N}(v_i)} (\mathbf{W}_u \mathbf{x}_j^{(t)}) \circ \lambda_u(\|\overrightarrow{\mathbf{r}}_{ji}\|) \circ \overrightarrow{\mathbf{x}}_j^{(t)} + (\mathbf{W}_v \mathbf{x}_j^{(t)}) \circ \lambda_v(\|\overrightarrow{\mathbf{r}}_{ji}\|) \circ \frac{\overrightarrow{\mathbf{r}}_{ji}}{\|\overrightarrow{\mathbf{r}}_{ji}\|} \end{align} ]

where ( \mathbf{W}h, \mathbf{W}u, \mathbf{W}_v ) are learnable matrices, ( \lambda ) functions are linear combinations of Gaussian radial basis functions, and ( \circ ) denotes the Hadamard product [28]. This approach maintains equivariance while avoiding computationally demanding tensor products used in other equivariant models.

Figure 1: E2GNN architecture illustrating the dual scalar-vector representation pathway and the four key message processing stages that maintain equivariance while ensuring computational efficiency.

Universal MLIPs: Performance Across Chemical and Dimensional Spaces

Benchmarking Phonon Properties

Harmonic phonon properties, derived from the second derivatives of the potential energy surface, provide a rigorous test for uMLIP accuracy near dynamically stable minima. Recent benchmarking of seven uMLIPs on approximately 10,000 non-magnetic semiconductors reveals significant performance variations [26].

Table 1: uMLIP Performance on Phonon Properties and Structural Relaxation

Model	Energy MAE (eV/atom)	Force Convergence Failure Rate (%)	Volume MAE (ų/atom)	Architecture Type
CHGNet	0.061	0.09	0.25	GNN with 3-body interactions
MatterSim-v1	0.035	0.10	0.21	M3GNet-based with active learning
M3GNet	0.035	0.21	0.24	Pioneering uMLIP with 3-body interactions
MACE-MP-0	0.026	0.21	0.20	Atomic cluster expansion
SevenNet-0	0.022	0.22	0.19	NequIP-based, equivariant
ORB	0.019	0.56	0.18	Smooth overlap of atomic positions
eqV2-M	0.016	0.85	0.16	Equivariant transformers

The benchmarking shows that while all uMLIPs achieve reasonable accuracy, models that predict forces as separate outputs (ORB and eqV2-M) rather than deriving them as energy gradients exhibit higher failure rates in geometry optimization, though they achieve lower energy errors [26]. This trade-off between accuracy and reliability must be considered when selecting models for PES exploration.

Dimensional Transferability

The ability of uMLIPs to describe systems across different dimensionalities—from 0D molecules to 3D bulk materials—is crucial for modeling real-world systems with mixed dimensionality, such as catalytic surfaces or interfaces. Recent benchmarking using the 0123D dataset (40,000 structures across dimensionalities) reveals that while most uMLIPs excel at 3D systems, accuracy degrades progressively for lower-dimensional structures [27] [29].

Table 2: Dimensional Transferability of uMLIPs (Position Error in Ã…)

Model	0D (Molecules)	1D (Nanowires)	2D (Layers)	3D (Bulk)	Training Data Size
eSEN-30m-oam	0.012	0.014	0.016	0.011	113M
ORB-v3-conservative	0.015	0.017	0.019	0.013	133M
MACE-mpa-0	0.018	0.020	0.022	0.015	12M
SevenNet-mf-ompa	0.019	0.021	0.023	0.016	113M
MatterSim-v1	0.023	0.025	0.027	0.019	17M
M3GNet	0.041	0.043	0.045	0.038	0.19M

The standout performer, eSEN (equivariant Smooth Energy Network), achieves remarkable consistency across all dimensionalities with atomic position errors of 0.01–0.02 Å and energy errors below 10 meV/atom, approaching quantum mechanical precision [27] [29]. The performance degradation in most models stems from training data biases heavily weighted toward 3D crystalline structures in databases like Materials Project and Alexandria [27].

High-Pressure Performance and Fine-Tuning

uMLIP performance under extreme pressure conditions (0-150 GPa) reveals significant limitations originating from fundamental gaps in training data rather than algorithmic constraints [30]. Benchmarking shows that while models excel at standard pressure, predictive accuracy deteriorates considerably with increasing pressure, with energy errors increasing from ~0.42 eV/atom to ~1.39 eV/atom for M3GNet at 150 GPa [30].

However, targeted fine-tuning on high-pressure configurations can substantially improve robustness. When fine-tuned on datasets containing high-pressure structures, models like MatterSim-ap-ft-0 and eSEN-ap-ft-0 show significantly restored predictive capability under high-pressure conditions [30]. This demonstrates that the limitations are data-centric rather than architectural, highlighting the importance of diverse training regimes for truly universal potentials.

Methodological Framework: Automated Potential Exploration

The autoplex Workflow

The autoplex framework implements automated iterative exploration and MLIP fitting through data-driven random structure searching (RSS), addressing the critical bottleneck of manual data generation and curation in MLIP development [2]. This approach unifies RSS with MLIP fitting, using gradually improved potential models to drive searches without relying on first-principles relaxations.

Figure 2: The autoplex automated workflow for iterative potential exploration and refinement, enabling robust MLIP development with minimal human intervention.

Application to Complex Systems

The autoplex framework has demonstrated wide-ranging capabilities across diverse systems. For elemental silicon, achieving target accuracy of 0.01 eV/atom required only ≈500 DFT single-point evaluations for highly symmetric diamond- and β-tin-type structures, though more complex metastable phases like oS24 silicon required a few thousand evaluations [2]. In the binary TiO₂ system, while common polymorphs (rutile, anatase) were readily captured, the bronze-type (B-) polymorph proved more challenging to learn, requiring additional iterations [2].

For full binary systems with multiple stoichiometric compositions (e.g., Ti–O system with Ti₂O₃, TiO, Ti₂O), achieving target accuracy required substantially more iterations due to the complex search space [2]. This highlights the framework's flexibility in handling varying stoichiometric compositions without additional user effort beyond input parameter adjustments.

Practical Implementation: The Researcher's Toolkit

Key Software and Framework Solutions

Table 3: Essential Research Reagents for Equivariant MLIP Development

Tool	Type	Primary Function	Key Features
autoplex	Software Framework	Automated MLIP exploration and fitting	Interoperability with atomate2, high-throughput RSS, minimal user intervention [2]
DeepChem Equivariant	Library	SE(3)-equivariant model implementation	Ready-to-use models (SE(3)-Transformer, TFNs), complete training pipelines, minimal DL background required [31]
e3nn	Library	Equivariant neural network infrastructure	Irreducible representations, spherical harmonics, tensor products [31]
DeePMD-kit	Software Package	Deep Potential Molecular Dynamics	Smooth neighbor descriptors, nonlinear atomic energy mapping, high performance [25]
MACE	Model Architecture	Higher order equivariant message passing	Excellent accuracy across dimensionalities, data efficiency [27]
0123D Dataset	Benchmark Data	Multi-dimensional performance evaluation	40,000 structures across 0D-3D, consistent computational parameters [29]
SW033291	SW033291, CAS:459147-39-8, MF:C21H20N2OS3, MW:412.6 g/mol	Chemical Reagent	Bench Chemicals
DL-Syringaresinol	(+)-Syringaresinol|High-Purity Lignan for Research		Bench Chemicals

Stability Testing Protocols

Compromised stability remains a critical challenge in MLIP deployment, particularly for molecular simulations in drug discovery. Rigorous testing protocols should include [32]:

• Normal Mode Analysis: Comparing vibrational frequencies against quantum mechanical references for simple benchmark molecules
• Gas Phase MD Stability: Assessing potential nonphysical behavior or simulation failures in isolated molecule dynamics
• Steric Clash Response: Evaluating model behavior at unphysically close interatomic distances
• Condensed Phase Reproduction: Testing ability to reproduce known liquid structures (e.g., radial distribution functions for water at ambient conditions)

These tests have revealed significant variations among public MLIPs, with some models exhibiting nonphysical additional energy minima in bond length/angle space, phase transitions to amorphous solids, or failure to maintain stable molecular dynamics simulations [32]. Only carefully trained models show better agreement with experimental data than simple molecular mechanics force fields like TIP3P [32].

The integration of equivariant architectures into universal machine learning interatomic potentials has fundamentally transformed the exploration of potential energy surfaces. Models like MACE, E2GNN, and eSEN demonstrate that embedding physical symmetries directly into network architectures enables unprecedented accuracy and data efficiency while maintaining computational practicality. Current benchmarks reveal that the best-performing uMLIPs now achieve errors approaching quantum mechanical accuracy (energy errors <10 meV/atom, position errors of 0.01–0.02 Å) across diverse dimensional regimes from molecules to bulk materials.

Nevertheless, significant challenges persist in achieving true universality. Performance gaps under high-pressure conditions, biases toward 3D structures in training data, and occasional instability in molecular dynamics simulations highlight the need for more diverse training datasets and robust architectural innovations. Frameworks like autoplex that automate the exploration and fitting process represent crucial infrastructure for addressing these limitations systematically.

As these advanced architectures continue to mature, they promise to accelerate materials discovery and drug development by enabling rapid, accurate sampling of potential energy surfaces at scales previously inaccessible to quantum mechanical methods. The integration of physically informed equivariant models with automated exploration frameworks marks a new era in computational materials science—one where the comprehensive mapping of complex potential energy surfaces becomes routine rather than exceptional.

In computational materials science and drug discovery, the high cost of acquiring labeled data is a fundamental bottleneck. Experimental synthesis and characterization often require expert knowledge, expensive equipment, and time-consuming procedures, while in silico methods like quantum-mechanical calculations demand substantial computational resources [33]. This challenge is particularly acute when exploring complex systems such as potential-energy surfaces (PESs), where understanding the relationship between atomic configuration and energy is crucial for predicting material properties and chemical behavior [2].

Two powerful, synergistic strategies have emerged to address this challenge: Active Learning (AL) and Random Structure Searching (RSS). Active learning is a supervised machine learning approach that strategically selects the most informative data points for labeling to optimize the learning process [34]. By iteratively selecting samples that maximize information gain, AL can achieve high model accuracy while minimizing labeling costs. Meanwhile, Random Structure Searching provides an efficient method for exploring configurational space by generating and evaluating diverse atomic structures [2]. When unified within automated frameworks, these approaches enable robust exploration and learning of complex scientific landscapes with unprecedented data efficiency.

This technical guide examines the core principles, methodologies, and applications of AL and RSS within machine learning research, with particular emphasis on their role in exploring potential energy surfaces for materials modeling and drug discovery.

Core Principles and Definitions

Active Learning Fundamentals

Active learning represents a paradigm shift from traditional passive learning, where models are trained on statically labeled datasets. Instead, AL operates through an iterative feedback process where the algorithm actively queries a human annotator or oracle for the most valuable data points to label [34] [35]. The primary objective is to minimize the labeled data required for training while maximizing model performance through intelligent data selection.

The theoretical foundation of active learning rests on the concept of sample informativeness – the potential of a data point to improve model parameters when incorporated into the training set. Formally, this can be expressed as selecting instances that maximize an acquisition function (a(x)) over the unlabeled pool (U):

[ x^* = \arg\max_{x \in U} a(x) ]

where (x^*) represents the most informative sample according to criteria such as prediction uncertainty, diversity, or expected model change [34] [33].

Random Structure Searching Fundamentals

Random Structure Searching is a computational approach for exploring the configuration space of atomic systems to identify stable and metastable structures. The original Ab Initio Random Structure Searching (AIRSS) approach involves generating random sensible atomic configurations, relaxing them using quantum-mechanical methods, and analyzing the resulting low-energy structures to map the potential-energy landscape [2].

Modern implementations combine RSS with machine-learned interatomic potentials (MLIPs) to dramatically reduce computational costs. By using gradually improved potential models to drive the searches, these approaches can explore configurational space without relying on expensive first-principles relaxations, requiring only limited single-point evaluations for refinement [2]. This synergy enables efficient navigation of high-dimensional potential-energy surfaces that would be prohibitively expensive to explore with quantum-mechanical methods alone.

Table 1: Key Comparative Overview of Active Learning and Random Structure Searching

Aspect	Active Learning (AL)	Random Structure Searching (RSS)
Primary Objective	Minimize labeling cost while maximizing model performance	Efficiently explore configurational space to identify stable structures
Core Methodology	Iterative querying of most informative samples for labeling	Generation and evaluation of diverse random atomic configurations
Key Metrics	Uncertainty, diversity, expected model change	Energy prediction error, structural diversity, discovery rate of stable phases
Domain Applications	Drug discovery, materials informatics, computer vision, NLP	Materials discovery, crystal structure prediction, molecular conformer search
Data Efficiency	Reduces required labeled samples by 60-70% [33]	Enables PES exploration with 70-95% fewer DFT calculations [2]

Methodologies and Experimental Protocols

Active Learning Query Strategies

The effectiveness of active learning hinges on the query strategy employed to select informative samples. Three primary categories of AL strategies have been developed, each with distinct mechanisms and applications:

Uncertainty Sampling: This approach selects instances where the model exhibits highest prediction uncertainty. Common techniques include:

• Least Confidence: Prefers instances with lowest maximum posterior probability
• Margin Sampling: Selects samples with smallest difference between top two class probabilities
• Entropy-based: Chooses instances with highest predictive entropy [34] [33]

Diversity Sampling: These methods aim to maximize the representativeness of selected samples by covering the input distribution. Approaches include:

• Cluster-based: Uses clustering algorithms to ensure selection from different data regions
• Core-set: Selects samples that approximate the entire dataset geometry
• Representative Sampling: Chooses instances similar to the overall data distribution [33]

Hybrid Methods: Combining uncertainty and diversity criteria often yields superior performance. The RD-GS strategy, for instance, integrates representativeness and diversity with a greedy search, showing particular effectiveness in early acquisition stages [33].

Stream-based Selective Sampling: In scenarios with continuous data generation, this approach processes instances sequentially, making immediate decisions about which samples to query for labeling based on informativeness measures [34].

Random Structure Searching Workflows

Modern RSS implementations, such as the autoplex framework, automate the exploration and fitting of potential-energy surfaces through structured workflows:

Initialization: The process begins with generating random sensible structures within defined compositional and geometrical constraints. For a binary system Ti-O, this would involve creating structures with varying stoichiometries and spatial arrangements [2].

Structure Evaluation: Initial structures are evaluated using a baseline model (either low-level quantum mechanics or preliminary MLIP) to obtain energy estimates. The autoplex framework uses Gaussian approximation potentials (GAP) for this purpose, leveraging their data efficiency [2].

Iterative Refinement: The key innovation in modern RSS is the iterative improvement of the MLIP using active learning:

• Selection: Identify structurally diverse or high-uncertainty configurations for high-level evaluation
• Labeling: Perform limited DFT single-point calculations on selected structures
• Update: Incorporate new data into the training set and refine the MLIP
• Exploration: Use the improved MLIP to guide further structure generation [2]

This cycle continues until target accuracy is achieved across relevant structural types, typically measured by root mean square error (RMSE) between predicted and reference energies.

Integrated AL-RSS Experimental Protocol

A comprehensive protocol for integrating active learning with random structure searching involves:

• Initial Data Collection:

  • Generate 100-500 random initial structures
  • Perform DFT single-point calculations (typically requiring 50-200 evaluations)
  • Train initial MLIP (e.g., GAP) on this dataset [2]

• Active Learning Loop:

  • Selection: Use query-by-committee or uncertainty sampling to identify 100-200 high-uncertainty structures
  • Labeling: Perform DFT single-point evaluations on selected structures
  • Update: Add newly labeled data to training set and retrain MLIP
  • Validation: Assess model on holdout set of known crystal structures [2]

• Convergence Criteria:

  • Target accuracy: RMSE < 0.01 eV/atom for energy predictions
  • Stable predictions across diverse structural types
  • Diminishing returns from additional data acquisition [2]

Table 2: Performance Benchmarks of Active Learning Strategies in Materials Science Regression Tasks [33]

AL Strategy	Principle	Early-Stage Performance (MAE)	Data Efficiency Gain	Best Use Cases
LCMD	Uncertainty-based	0.18	65%	Small data budgets (<30% of total)
Tree-based-R	Uncertainty-based	0.19	63%	High-dimensional feature spaces
RD-GS	Diversity-Representativeness	0.20	60%	Initial exploration phases
GSx	Geometry-only	0.25	45%	Baseline comparison
EGAL	Geometry-only	0.26	42%	Simple feature spaces
Random	Baseline	0.28	0%	Control experiments

Applications in Potential Energy Surface Exploration

Materials Discovery and Optimization

The AL-RSS combination has demonstrated remarkable success in materials discovery applications. In the titanium-oxygen system, automated exploration with autoplex enabled accurate modeling of multiple phases with varied stoichiometric compositions (Ti2O3, TiO, Ti2O) [2]. The framework achieved quantum-mechanical accuracy (RMSE < 0.01 eV/atom) for structurally diverse polymorphs including rutile, anatase, and bronze-type TiO2, with progressive improvement as more structures were incorporated [2].

For phase-change memory materials, the AL-RSS approach efficiently navigated complex energy landscapes to identify metastable phases relevant to device operation. Similarly, applications to SiO2 and water systems demonstrated robust parameterization for both crystalline and liquid phases, highlighting the transferability of the methodology across different bonding environments [2] [18].

Drug Discovery and Molecular Optimization

In pharmaceutical applications, active learning addresses the challenge of navigating vast chemical spaces with limited experimental data. AL strategies have been successfully applied to:

Compound-Target Interaction Prediction: Active learning efficiently identifies valuable data within vast chemical space, even with limited labeled data, making it particularly valuable for predicting compound-target interactions where experimental binding data is scarce [35].

ADMET Property Optimization: Batch active learning methods have shown significant improvements in predicting absorption, distribution, metabolism, excretion, and toxicity properties. Novel batch selection methods like COVDROP and COVLAP, which maximize joint entropy across batches, have demonstrated 30-50% reduction in experimental requirements for achieving target model performance [36].

Molecular Generation and Optimization: Frameworks like TRACER integrate molecular property optimization with synthetic pathway generation using reinforcement learning guided by active learning. This approach successfully generated compounds with high predicted activity against DRD2, AKT1, and CXCR4 targets while maintaining synthetic feasibility [37].

Table 3: Performance of Active Learning in Drug Discovery Applications [36]

Dataset/Property	Standard Approach RMSE	AL-Enhanced RMSE	Experimental Reduction
Aqueous Solubility	0.98 (at 20% data)	0.72 (at 20% data)	40%
Cell Permeability (Caco-2)	0.45 (at 30% data)	0.32 (at 30% data)	35%
Lipophilicity	0.64 (at 25% data)	0.51 (at 25% data)	30%
Plasma Protein Binding	1.2 (at 40% data)	0.87 (at 40% data)	45%

The Scientist's Toolkit: Essential Research Reagents

Implementing effective AL-RSS workflows requires specialized computational tools and frameworks. Key resources include:

autoplex: An automated framework for exploration and fitting of potential-energy surfaces, designed for interoperability with existing software architectures and high-throughput computation on HPC systems [2] [18].

Gaussian Approximation Potential (GAP): A machine learning interatomic potential framework that enables data-efficient modeling of atomic interactions, particularly valuable for RSS applications [2].

DeepChem: An open-source toolkit for drug discovery that provides implementations of various active learning strategies, including novel batch selection methods [36].

Atomate2: A materials science workflow framework that provides foundational infrastructure for automated computation and data management, serving as a core component for systems like autoplex [2].

BAIT (Bayesian Active Learning by Disagreement): A batch active learning method that uses Fisher information to optimally select samples that maximize information gain, particularly effective with neural network models [36].

Monte Carlo Dropout: A practical approach for uncertainty estimation in deep learning models, enabling uncertainty-based active learning without requiring multiple model instances [36].

Workflow Visualization

Integrated AL-RSS Workflow Diagram: This visualization illustrates the synergistic relationship between Active Learning and Random Structure Searching in exploring potential energy surfaces. The workflow begins with initialization, proceeds through iterative refinement cycles where each component informs the other, and concludes when model convergence criteria are satisfied.

Future Directions and Challenges

Despite significant advances, several challenges remain in the widespread adoption of AL-RSS methodologies. Reproducibility and inconsistent methodologies across studies present barriers to comparative evaluation [38]. As automated frameworks mature, standardization of benchmarks and evaluation metrics will be crucial for community progress.

The integration of AL with foundational or pre-trained models represents a promising direction. Recent trends toward "foundational MLIPs" pre-trained on diverse chemical spaces could be combined with active learning for efficient fine-tuning to specific systems [2]. Similarly, in drug discovery, transfer learning from large chemical databases enhanced with AL for target-specific optimization shows considerable potential [35] [36].

Technical challenges in uncertainty quantification persist, particularly for regression tasks common in materials science [33]. Improved uncertainty estimation methods that remain robust under changing model architectures during AutoML optimization are an active area of research.

Finally, extending these methodologies to more complex systems—including surfaces, interfaces, and reaction pathways—represents an important frontier for future work [2]. As frameworks become more sophisticated and computational resources grow, AL-RSS approaches will likely play an increasingly central role in computational materials science and drug discovery.

Computational chemistry relies on high-level quantum mechanics methods, such as coupled cluster theory with single, double, and perturbative triple excitations (CCSD(T)), to achieve accurate results for molecular properties and reaction barriers. These methods provide the "gold standard" of quantum chemistry accuracy but come with prohibitive computational costs that scale severely with system size [39]. This creates a significant challenge for studying complex reactions and large molecular systems, including those relevant to drug discovery and catalyst design [40] [41].

Density functional theory (DFT) and other low-level quantum methods offer a computationally feasible alternative for larger systems but often lack the required accuracy for reliable predictions [39]. This accuracy gap is particularly problematic for calculating activation energies in complex reactions, where small energy differences can dramatically impact predicted reaction rates and selectivity [42].

Δ-machine learning (Δ-ML) has emerged as a powerful framework to bridge this computational trade-off. The core concept involves learning the difference (Δ) between high-level and low-level calculations, rather than learning the target property directly [43]. This approach enables researchers to combine the computational efficiency of low-level methods with the accuracy of high-level theories, making near-CCSD(T) quality calculations feasible for complex molecular systems [39].

Theoretical Foundation of Δ-Machine Learning

Core Mathematical Framework

The Δ-machine learning approach is built on a simple but powerful mathematical premise:

V_Final = V_LL + Δ_HL-LL

Where:

• V_Final represents the final predicted property at high-level accuracy
• V_LL represents the property calculated using a low-level method
• Δ_HL-LL represents the machine-learned correction term [43]

This formulation can be applied to various molecular properties, including potential energy surfaces (PES), force fields, and activation energies [39] [42]. The Δ-ML model is trained to predict the difference between reference high-level calculations and corresponding low-level computations, typically using a relatively small set of high-level reference data [43].

Key Methodological Variations

Several methodological implementations of Δ-machine learning have been developed, each with distinct advantages:

• Potential Energy Surface Refinement: Using permutationally invariant polynomials (PIP) to fit high-dimensional PESs, where Δ-ML corrects a DFT-based PES to near-CCSD(T) accuracy [43]
• Force Field Enhancement: Applying many-body corrections to polarizable force field potentials using CCSD(T) datasets for water clusters [39]
• Activation Energy Prediction: Employing graph neural networks to predict differences between semiempirical quantum mechanics and CCSD(T)-F12a activation energies [42]

Table 1: Comparison of Δ-ML Methodologies and Their Applications

Methodology	Target Property	Low-Level Method	High-Level Method	System Demonstrated
PIP-based Δ-ML	Potential Energy Surface	DFT/B3LYP/6-31+G(d)	CCSD(T)-F12a	CH₄, H₃O⁺, N-methylacetamide [43]
Graph Neural Network Δ-ML	Activation Energy	Semiempirical QM	CCSD(T)-F12a	Diverse reaction database [42]
Many-body Correction Δ-ML	Force Fields	TTM2.1 water potential	CCSD(T)	Water clusters (2-b, 3-b, 4-b) [39]
BAY-549	BAY-549, CAS:867017-68-3, MF:C18H13ClF2N6O, MW:402.8 g/mol	Chemical Reagent	Bench Chemicals
WF-536	WF-536, CAS:539857-64-2, MF:C14H16ClN3O, MW:277.75 g/mol	Chemical Reagent	Bench Chemicals

Implementation Protocols: From Theory to Practice

Workflow for Potential Energy Surface Generation

The complete workflow for developing a Δ-ML refined potential energy surface involves multiple stages of computational chemistry and machine learning:

Detailed Computational Methodology

Step 1: Low-Level Data Generation

• Perform geometry optimizations and single-point energy calculations using efficient but approximate methods (DFT with functionals like B3LYP, or semiempirical quantum mechanics)
• Calculate energy gradients (forces) for molecular dynamics sampling
• For PES development, sample configurations using molecular dynamics trajectories at relevant temperatures [43]

Step 2: High-Level Reference Calculations

• Select a representative subset of configurations for high-level calculation (typically 200-5000 points depending on system size)
• Perform single-point energy calculations at the CCSD(T) level or similar high-accuracy methods
• For 15-atom systems like tropolone, consider specialized approaches like molecular tailoring or local CCSD(T) to reduce computational cost [39]

Step 3: Feature Engineering and Representation

• For molecular systems: use permutationally invariant polynomials (PIPs) to maintain physical constraints [43]
• For reaction systems: implement condensed graph of reaction (CGR) representation combining reactants and products [42]
• Utilize graph neural networks (e.g., directed message passing neural networks) to automatically learn relevant features from molecular structure [42]

Step 4: Model Training and Validation

• Train machine learning models (PIP fits, neural networks, or graph networks) to predict Δ = E_HL - E_LL
• Apply k-fold cross-validation with stratified sampling to ensure representative coverage of chemical space
• Validate against hold-out sets of high-level calculations not used in training
• For PES applications, validate against experimental spectroscopic data or reaction rates when available [43]

Performance Comparison of Enhancement Methods

Recent systematic studies have compared Δ-machine learning against other approaches for enhancing low-level computational data:

Table 2: Performance Comparison of Machine Learning Enhancement Methods for Activation Energy Prediction [42]

Method	Description	Key Advantage	Limitations	Data Efficiency
Δ-Learning	Predicts difference between low and high-level	Highest accuracy; matches full data set performance with only 20-30% of high-level data	Requires transition state searches during application	Excellent
Transfer Learning	Pretrains on large low-level datasets, fine-tunes on high-level	Leverages abundant low-level data	Performance depends on distribution match between datasets	Moderate
Feature Engineering	Adds computed molecular properties as input features	Modest gains without architectural changes	Limited improvement for complex reactions	Low

Applications in Catalysis and Drug Discovery

Catalyst Design and Reaction Engineering

In catalysis research, Δ-ML enables accurate exploration of complex potential energy surfaces that dictate catalyst selectivity and reactivity [41]. This approach is particularly valuable for:

• Transition State Energy Prediction: Accurately determining activation barriers for catalytic reactions without the computational cost of full CCSD(T) transition state searches for all pathways
• High-Throughput Screening: Rapidly evaluating thousands of potential catalyst materials with near-CCSD(T) accuracy
• Reaction Mechanism Elucidation: Mapping complete reaction networks with accurate energetics for complex catalytic processes [41]

The method has been successfully applied to heterogeneous catalysis systems, where it helps identify correlations between microscopic catalyst structure and performance metrics like turn-over frequency and selectivity [41].

Pharmaceutical Drug Development

In drug discovery, Δ-ML accelerates multiple stages of the development pipeline:

• Molecular Modeling and Drug Design: Improving the accuracy of binding affinity predictions between drug candidates and target proteins [40]
• Virtual Screening: Enhancing the selection of promising drug candidates from large chemical libraries by providing more reliable energy calculations [40] [44]
• Activation Energy Prediction: For metabolic pathway analysis, accurately predicting activation energies for enzyme-catalyzed reactions of drug candidates [42]

The implementation of Δ-ML in pharmaceutical research addresses key bottlenecks in traditional drug discovery, including high failure rates, time-intensive processes, and astronomical costs that can reach $2.6 billion per approved drug [44].

Research Reagent Solutions: Software and Datasets

Table 3: Essential Tools for Δ-Machine Learning Implementation

Tool/Category	Specific Examples	Function/Purpose	Application Context
Quantum Chemistry Software	Gaussian, PySCF, FIREBALL, ORCA	Perform low and high-level quantum calculations	Generate reference data for Δ-ML training [41]
Machine Learning Libraries	Chemprop, TensorFlow, PyTorch	Implement neural network models	Develop Δ-ML correction models [42]
Specialized Δ-ML Tools	PIP package, q-AQUA water potential	Domain-specific Δ-ML implementations	Potential energy surface generation [39] [43]
Reaction Datasets	Spiekermann et al. database, Grambow et al. dataset	Provide curated reaction energies and barriers	Benchmark Δ-ML performance [42]
Molecular Representation	RDKit, SMILES, Condensed Graph of Reaction (CGR)	Convert molecular structures to machine-readable features	Preprocess input data for graph neural networks [42]

Comparative Methodologies and Performance

Relationship to Other Machine Learning Approaches

Δ-machine learning differs fundamentally from other data enhancement strategies in computational chemistry:

Direct Learning approaches train models to predict properties directly from molecular structure, requiring large amounts of high-quality training data. In contrast, Δ-ML explicitly leverages the physical knowledge embedded in low-level calculations and only learns the correction term [42].

Quantitative Performance Metrics

Recent systematic evaluations demonstrate the superior data efficiency of Δ-ML:

• For activation energy prediction, Δ-ML trained with just 20-30% of high-level data matched or exceeded the performance of other methods trained with the full dataset [42]
• In PES development for molecules like N-methylacetamide, Δ-ML achieved CCSD(T) quality with only 4,696 CCSD(T) energies for a 12-atom system [43]
• For water cluster interactions, Δ-ML enabled development of fully ab initio potentials (q-AQUA) that accurately reproduce CCSD(T) level interactions [39]

Future Perspectives and Challenges

The continued development of Δ-machine learning faces several important frontiers:

Data Quality and Availability: As with all machine learning approaches, Δ-ML depends on the quality and representativeness of training data. Developing standardized datasets and benchmarking protocols remains crucial for advancing the field [42].

Transferability and Generalization: Ensuring that Δ-ML models trained on specific chemical systems can generalize to novel compounds and reactions is an ongoing challenge that requires careful feature engineering and model architecture design [41].

Integration with High-Throughput Workflows: Future developments will focus on streamlining Δ-ML implementation within automated computational workflows, making high-accuracy calculations more accessible to non-specialists [40] [41].

Methodological Hybridization: Combining Δ-ML with other enhancement strategies like transfer learning and advanced feature engineering may yield further improvements in accuracy and efficiency [42].

As computational resources grow and algorithms improve, Δ-machine learning is poised to become an increasingly standard component of the computational chemist's toolkit, particularly for drug discovery and catalyst design where accurate energetics are essential for reliable predictions.

Understanding Potential Energy Surfaces (PES) is fundamental to pharmaceutical research, as it enables scientists to identify optimal molecular conformations and transition states during chemical reactions [45]. A thorough grasp of the PES provides crucial information on the intricate interactions between drug molecules and their receptor sites at the atomic level, where binding strength greatly influences therapeutic efficacy [45]. The dynamic nature of biomolecules means that proteins sample many conformational states, both open and closed, which are selectively stabilized by ligand binding [46]. Molecular dynamics (MD) simulations and machine learning (ML) approaches have emerged as powerful tools for exploring these complex energy landscapes, moving beyond static structural models to capture the continuous jiggling and wiggling of atoms that characterizes biological systems [46].

Computational Methods for Exploring Biomolecular Energy Landscapes

Traditional Simulation Approaches

Molecular Dynamics (MD) Simulations approximate atomic motions using Newtonian physics, representing atoms and bonds as simple spheres connected by virtual springs [46]. These simulations calculate forces from interactions between bonded and non-bonded atoms, with chemical bonds modeled using virtual springs, dihedral angles modeled using sinusoidal functions, and non-bonded forces arising from van der Waals and electrostatic interactions [46]. Despite their utility, traditional MD simulations face significant limitations: they are computationally intensive (with microsecond simulations taking months to complete), use approximate force fields that require further refinement, and poorly model quantum effects crucial for understanding chemical reactions [46].

Quantum Mechanics (QM) Methods like density functional theory (DFT) provide higher accuracy but at substantially greater computational cost, making them impractical for large biomolecular systems [45]. The wB97X/6-31G(d) level of theory has gained popularity for studying ground states of various compounds due to its computational efficiency and accuracy [45].

Machine Learning-Enhanced Approaches

Recent advances have integrated machine learning to overcome limitations of traditional methods. Neural Network Potentials (NNPs) map atomic structure to potential energy, significantly improving computational efficiency compared to traditional PES methods while maintaining high accuracy [47]. The ANI (ANAKIN-ME) model represents a accurate deep learning-based neural network potential method that utilizes a modified version of Behler-Parrinello symmetry functions to build atomic environment vectors as molecular representations [45]. Frameworks like autoplex implement automated exploration and MLIP fitting through data-driven random structure searching, enabling high-throughput potential development [2].

Table 1: Comparison of Computational Methods for PES Exploration

Method	Computational Cost	Accuracy	Key Applications	Limitations
Classical MD [46]	Moderate to High	Moderate	Protein folding, ligand binding, conformational changes	Cannot model chemical reactions; force field approximations
QM/DFT [45]	Very High	High	Electronic structure, reaction mechanisms	Limited to small systems; computationally intensive
Neural Network Potentials (e.g., ANI-1x) [45] [47]	Low to Moderate	High to Very High	Large-scale simulations with quantum accuracy	Training data quality dependency; potential overfitting
Automated Frameworks (e.g., autoplex, ARplorer) [2] [47]	Variable	High	High-throughput PES exploration, reaction pathway prediction	Implementation complexity; system-specific optimization needed

Machine Learning Framework for Reaction Pathway Exploration

Integrated Workflow Design

The ARplorer program exemplifies modern approaches to reaction pathway exploration by combining quantum mechanics with rule-based methodologies, underpinned by Large Language Model-assisted chemical logic [47]. This program operates on a recursive algorithm with three key steps: (1) identifying active sites and potential bond-breaking locations to set up multiple input molecular structures, (2) optimizing molecular structure through iterative transition state searches using active-learning sampling, and (3) performing Intrinsic Reaction Coordinate analysis to derive new reaction pathways [47]. The flexibility to switch between computational methods (e.g., GFN2-xTB for quick screening and DFT for precise calculations) makes this approach particularly versatile for drug discovery applications [47].

LLM-Guided Chemical Logic Implementation

A particularly innovative aspect of modern PES exploration is the integration of Large Language Models to encode chemical knowledge [47]. The chemical logic in ARplorer is built from two complementary components: pre-generated general chemical logic derived from scientific literature, and system-specific chemical logic generated by specialized LLMs [47]. General chemical logic generation begins by processing and indexing prescreened data sources (books, databases, research articles) to form a comprehensive chemical knowledge base, which is then refined into general SMARTS patterns [47]. For system-specific rules, reaction systems are converted into SMILES format, enabling specialized LLMs to generate tailored chemical logic and SMARTS patterns [47].

Experimental Protocols and Validation

Performance Benchmarks

Machine learning approaches have demonstrated remarkable accuracy in predicting molecular properties. In a study on the Resveratrol molecule, the ANI-1x neural network potential predicted electronic energy with comparable performance to DFT calculations at the wB97X/6-31G(d) level of theory [45]. The ANI-1x model demonstrated the ability to correctly recognize differences between aromatic and nonaromatic carbon-carbon bond lengths in molecular structures, accurately predicting the chemical environment of double bonds [45]. For example, while C3-C4 and C4-C5 aromatic bond lengths were calculated at 1.39422 and 1.39830 Ã…, respectively, the C5-C6 and C7-C8 nonaromatic bond lengths were correctly identified as 1.48132 and 1.47907 Ã… [45].

Table 2: ANI-1x Performance on Resveratrol Molecular Structure [45]

Parameter	ANI-1x Prediction	DFT Reference (wB97X/6-31G(d))	Deviation
Electronic Energy (kcal/mol)	-480,773.2	-480,772.4	0.8 kcal/mol
C3-C4 Aromatic Bond Length (Ã…)	1.39422	1.39447	0.00025 Ã…
C5-C6 Nonaromatic Bond Length (Ã…)	1.48132	1.48125	0.00007 Ã…
C6-C7 Double Bond Length (Ã…)	1.33782	1.33795	0.00013 Ã…
Vibrational Frequency RMSE	43.0 cm⁻¹	Reference	43.0 cm⁻¹

Automated Framework Validation

The autoplex framework has been validated across diverse systems, from elemental silicon to complex binary titanium-oxygen systems [2]. In testing, the approach achieved accuracies on the order of 0.01 eV/atom for silicon allotropes with only a few hundred DFT single-point evaluations [2]. For more complex systems like TiO₂ polymorphs, accurate description of common forms (rutile and anatase) required minimal computational effort, while the bronze-type polymorph presented greater challenges for the learning algorithm [2]. This framework demonstrates particular strength in handling varying stoichiometric compositions without substantially greater user effort—only a change in input parameters for random structure searching is required [2].

Research Reagent Solutions: Computational Tools for Drug Discovery

Table 3: Essential Computational Tools for Biomolecular Simulation and PES Exploration

Tool/Platform	Type	Primary Function	Application in Drug Discovery
ANI-1x/ANI-2x [45] [48]	Neural Network Potential	Accelerated quantum-mechanical calculations	Predicting molecular energies and structures with DFT-level accuracy
ARplorer [47]	Automated Exploration Program	Reaction pathway mapping using QM and rule-based methods	Multi-step reaction mechanism elucidation for drug metabolism studies
autoplex [2]	Automated MLIP Framework	High-throughput potential energy surface exploration	Rapid screening of drug-receptor binding conformations
Gaussian 09 [47]	Quantum Chemistry Software	Electronic structure modeling	Reference calculations for reaction barrier heights
GFN2-xTB [47]	Semiempirical Method	Fast PES generation and large-scale screening	Preliminary screening of reaction pathways and conformers
AMBER/CHARMM [46]	Molecular Dynamics Force Field	Biomolecular simulation	Protein-ligand binding dynamics and conformational sampling

The integration of machine learning with traditional simulation methods represents a paradigm shift in computational drug discovery. ML-enhanced approaches like neural network potentials and automated exploration frameworks are addressing fundamental challenges in molecular simulations, particularly the competing demands of computational efficiency and quantum-mechanical accuracy [45] [2]. The incorporation of large language models to encode chemical logic further enhances the capability of these systems to navigate complex reaction pathways relevant to pharmaceutical development [47].

As specialized hardware like graphics processing units (GPUs) and application-specific integrated circuits (ASICs) continue to evolve, alongside algorithmic advances in active learning and enhanced sampling, we anticipate increasingly robust and automated pipelines for biomolecular simulation [48]. These developments will enable more comprehensive exploration of potential energy surfaces, ultimately accelerating the identification and optimization of novel therapeutic compounds through deeper understanding of reaction mechanisms and drug-target interactions.

Overcoming Practical Hurdles: Data Fidelity, Model Generalizability, and Computational Efficiency

In the field of computational chemistry and materials science, the accurate exploration of potential energy surfaces (PES) is fundamental to understanding and predicting molecular behavior, chemical reactions, and material properties. Machine learning (ML) has emerged as a transformative tool for constructing highly accurate and computationally efficient interatomic potentials, known as machine learning interatomic potentials (MLIPs). These models promise to deliver density functional theory (DFT)-level accuracy at a fraction of the computational cost, potentially unlocking the simulation of scientifically relevant molecular systems and reactions of real-world complexity that have always been out of reach [49]. However, the performance and reliability of these ML models are profoundly constrained by a fundamental dilemma: the tension between the quantity and quality of training data. This whitepaper examines this core challenge within the context of PES research, providing researchers and drug development professionals with methodologies and frameworks for sourcing and generating reliable training sets that balance these competing demands.

The prevailing paradigm in ML model development has often emphasized dataset scale, operating under the assumption that more data invariably leads to better models. While this holds some truth, reliance on insufficiently diverse data, particularly data limited to DFT relaxation trajectories, fundamentally constrains model accuracy and generalizability [50]. The adage "garbage in, garbage out" remains painfully true; robust machine learning models can be crippled when trained on inadequate, inaccurate, or irrelevant data [51]. The consequences of poor data quality include unphysical predictions, failure to simulate reactive events, and ultimately, unreliable scientific conclusions. This paper argues that a strategic, quality-first approach to data generation, emphasizing comprehensive sampling of configurational space and robust validation, is paramount for advancing the state-of-the-art in ML-driven PES exploration.

The Quantitative Landscape: Modern PES Datasets

The computational chemistry community has recently witnessed the release of several landmark datasets that illustrate the evolving strategies for balancing data quantity with quality. The table below summarizes key characteristics of recent major datasets, highlighting their different approaches to this challenge.

Table 1: Comparison of Recent Major Datasets for Machine-Learned Interatomic Potentials

Dataset Name	Size (Structures)	Sampling Strategy	Chemical Diversity	Key Properties
MatPES [50]	~400,000	Careful selection from 281 million MD snapshots (16B atomic environments)	Foundational across periodic table	Energies, Forces (r²SCAN functional)
Open Molecules 2025 (OMol25) [49]	>100 Million	DFT simulations on curated content from past datasets and new focus areas (biomolecules, electrolytes, metal complexes)	Heavy elements, metals, biomolecules, electrolytes	3D molecular snapshots, Energies, Forces
QCML Dataset [52]	33.5M (DFT) 14.7B (Semi-empirical)	Systematic conformer search and normal mode sampling from 17.2M chemical graphs	Small molecules (≤8 heavy atoms), large fraction of periodic table	Energies, Forces, Multipole moments, Kohn-Sham matrices

These datasets demonstrate a shift from purely quantity-driven efforts to more nuanced strategies. MatPES, while modest in total final size, is curated from an enormous pool of candidate structures, emphasizing data quality over raw quantity [50]. In contrast, OMol25 achieves both scale and diversity, costing six billion CPU hours—over ten times more than any previous dataset—to generate over 100 million 3D molecular snapshots that are substantially more complex than past efforts, with up to 350 atoms including challenging heavy elements and metals [49]. The QCML dataset employs a hierarchical strategy, using extensive semi-empirical calculations to guide a smaller but highly valuable set of DFT calculations [52].

Methodological Framework for High-Quality Data Generation

Active Learning and Negative Design

A critical advancement in generating training data for PES is the move from static datasets to dynamic, intelligent sampling via active learning workflows. This is particularly crucial for modeling complex, reactive chemistry like hydrogen combustion, where traditional reliance on chemical intuition can lead to incomplete PES descriptions and flawed models [53].

Active learning frameworks iteratively improve the MLIP by identifying and incorporating new, informative data points. The workflow often employs a query-by-committee approach, where multiple ML models are trained on the same initial data. When these models disagree significantly on the prediction for a new configuration, it signals high uncertainty, and that configuration is then selected for accurate (and expensive) ab initio calculation. The newly labeled data is added to the training set, and the models are retrained, progressively improving their accuracy and coverage [53].

Complementing this, the "negative design" strategy uses enhanced sampling methods, such as metadynamics, to actively explore high-energy or unphysical regions of the PES that might be overlooked by standard molecular dynamics but are critical for capturing transition states and reaction pathways. This helps create a more complete and robust ML model that avoids unforeseen failures [53]. The following diagram illustrates this integrated workflow.

Hierarchical Data Generation and Conformational Sampling

For comprehensive coverage of chemical space, a hierarchical data generation strategy is highly effective. The QCML dataset exemplifies this approach, organizing data on three levels: chemical graphs, molecular conformations, and quantum chemical calculation results [52].

The process begins with sourcing and generating diverse chemical graphs, which are representations of molecular connectivity. These graphs are then used to generate a wide array of 3D conformations through systematic conformer search and normal mode sampling at various temperatures, ensuring coverage of both equilibrium and off-equilibrium structures essential for training force fields. Finally, high-level quantum chemical calculations are performed on a strategically selected subset of these conformations [52]. This method ensures that the resulting dataset is both broad and deep, covering a vast chemical space without sacrificing the accuracy of the reference data.

The Scientist's Toolkit: Essential Research Reagents and Infrastructure

Building reliable training sets for PES exploration requires a suite of computational "research reagents." The table below details key resources, their functions, and considerations for their use.

Table 2: Essential Research Reagents for PES Data Generation and Model Training

Tool Category	Specific Examples	Function & Application	Technical Notes
Reference Quantum Chemistry Methods	Density Functional Theory (DFT), r²SCAN functional [50]	Provides high-accuracy reference energies and forces for training; the "ground truth."	r²SCAN offers improved bonding description; DFT balances accuracy and cost.
Active Learning & Sampling Tools	Metadynamics [53], PLUMED [53]	Enhances sampling of rare events and high-energy regions for negative design.	Critical for exploring reaction pathways and transition states beyond equilibrium MD.
Dataset Repositories	OMol25 [49], MatPES [50], QCML [52]	Pre-computed datasets for initial model training or transfer learning.	Assess dataset's chemical diversity, property coverage, and level of theory.
Model Evaluation Suites	OMol25 Evaluations [49]	Standardized benchmarks to measure and track MLIP performance on specific tasks.	Enables objective model comparison and builds trust in ML predictions for complex chemistry.
High-Performance Computing (HPC)	CPU/GPU Clusters, Cloud Computing	Provides the computational infrastructure for DFT calculations and ML model training.	OMol25 cost ~6B CPU hours; Cloud costs require FinOps for optimization [49] [54].
UCB-35440	UCB-35440|Poorly Soluble Research Compound		Bench Chemicals
BAY-320	BAY-320, CAS:288250-47-5, MF:C27H29ClN6O2, MW:505.0 g/mol	Chemical Reagent	Bench Chemicals

The effectiveness of these tools is interdependent. For instance, the choice of reference quantum chemistry method (e.g., the r²SCAN functional for its improved bonding descriptions [50]) directly impacts the quality of the training data. Similarly, the scale of computing required, as exemplified by the six billion CPU hours needed for OMol25, necessitates robust HPC infrastructure and careful cost management through practices like FinOps to avoid budget overruns [49] [54].

A Protocol for Data Generation and Model Validation

This section outlines a detailed, actionable protocol for generating a high-quality training set for a MLIP targeting a specific chemical reaction, such as hydrogen combustion [53].

Phase 1: System Setup and Initial Data Acquisition

• Define System and Goals: Clearly delineate the chemical system, relevant elements, and the range of pressures and temperatures of interest. For hydrogen combustion, this involves defining the stoichiometry and reaction conditions for the combustion process.
• Assemble Initial Dataset: Compile an initial set of structures from existing sources, such as reactant and product geometries, known intermediate states, and transition states from literature or previous calculations. This serves as the foundational dataset.
• Perform Initial Ab Initio Calculations: Calculate high-fidelity reference energies and forces for all structures in the initial dataset using an appropriate level of theory (e.g., ωB97X-V/def2-TZVP). This establishes the initial training data.

Phase 2: Active Learning Cycle

• Train Committee of Models: Train multiple MLIPs (the "committee") on the current training set.
• Run Enhanced Sampling Molecular Dynamics: Launch molecular dynamics simulations, preferably biased with metadynamics, to explore the PES. The collective variables for metadynamics should be chosen to drive the system along suspected reaction pathways.
• Identify and Label Uncertain Configurations: For each new configuration visited during MD, query the committee of models. If the model predictions for energy/forces diverge beyond a predefined threshold (e.g., a query-by-committee disagreement metric), select that configuration for ab initio calculation.
• Retrain and Iterate: Incorporate the new ab initio-labeled data into the training set. Retrain the ML models and repeat the cycle from Step 2 until the model performance converges and no further high-uncertainty regions are discovered during a full MD run.

Phase 3: Validation and Benchmarking

• Independent Benchmarking: Evaluate the final model's performance on a held-out test set of configurations that were not included in the active learning loop.
• Challenging Evaluations: Use dedicated evaluation sets, like those provided for OMol25, to test the model on specific, challenging tasks such as bond breaking/formation, and predicting properties of molecular complexes with variable charges and spins [49].
• Free Energy Calculation: The ultimate test is the model's ability to reproduce experimental observables. Use the MLIP to run long-timescale MD for calculating the free-energy change of the reaction transition-state mechanism and compare against experimental or high-level theoretical benchmarks [53].

The dilemma between data quality and quantity in training set generation for PES exploration is not resolved by choosing one over the other, but through strategic integration. The future lies in systematic, intelligent data acquisition that prioritizes diversity, uncertainty-driven sampling, and rigorous validation. As evidenced by recent large-scale community efforts, the focus is shifting from merely accumulating data to curating high-quality, chemically diverse datasets that enable the development of foundational, generalizable, and reliable MLIPs. By adopting the methodologies and frameworks outlined in this whitepaper—active learning, negative design, hierarchical generation, and robust validation—researchers and drug development professionals can build trustworthy ML models that truly unlock the power of atomistic simulation for materials discovery and design.

The exploration of potential-energy surfaces (PES) is fundamental to advancements in materials modelling and drug discovery, enabling large-scale atomistic simulations with quantum-mechanical accuracy [2]. Machine learning interatomic potentials (MLIPs) have become the method of choice for this task, but their development hinges on high-quality training data that comprehensively represents the relevant chemical space [2]. A critical challenge emerges when training data lacks uniform coverage of biomolecular structures, creating a dimensionality bias that severely limits model generalizability [55]. This coverage bias represents a significant pitfall, as models trained on non-uniform data may perform well within their restricted training domain but fail to predict properties accurately for novel molecular structures outside this domain [55] [56].

The problem is analogous to spatial bias in geographical analysis, where models trained on data from one location fail to generalize to other regions [55]. In molecular machine learning, this manifests when a model trained predominantly on lipids is applied to flavonoids with no reasonable expectation of success [55]. Understanding and mitigating this bias is therefore crucial for developing reliable MLIPs and molecular property predictors that can accurately navigate and explore potential energy surfaces across diverse chemical spaces.

The Coverage Bias Problem in Molecular Machine Learning

Fundamental Concepts and Definitions

Coverage bias in molecular machine learning refers to the non-uniform representation of chemical structures in training datasets, which fails to adequately sample the true distribution of known biomolecular structures [55]. This bias stems from practical constraints in data collection, where the availability of compounds is governed by factors such as difficulty of chemical synthesis, commercial availability of precursor compounds, and associated costs [55]. The lower the availability of a compound, the higher its price, and the less likely it is to be included in large-scale datasets—creating a systematic gap in chemical space coverage.

The domain of applicability defines the region of chemical space where a model's predictions can be trusted, bounded by the chemical diversity present in its training data [55]. When models are applied outside this domain, predictions become unreliable. The Maximum Common Edge Subgraph (MCES) distance provides a chemically intuitive measure of structural similarity that aligns well with chemical intuition, serving as a valuable metric for assessing molecular coverage [55].

Quantitative Evidence of Coverage Gaps

Recent research analyzing 14 molecular structure databases containing 718,097 biomolecular structures has revealed significant coverage gaps in widely-used datasets [55]. By implementing a computationally efficient approach combining Integer Linear Programming and heuristic bounds to compute MCES distances, researchers found that many popular training datasets lack uniform coverage of biomolecular structures, directly limiting the predictive power of models trained on them [55].

Table 1: Analysis of Biomolecular Structure Coverage in Combined Databases

Analysis Metric	Finding	Implication
Database Size	718,097 biomolecular structures from 14 databases	Proxy for the "universe of small molecules of biological interest"
Sampling Analysis	20,000 structures uniformly subsampled for analysis	Computational constraints necessitate strategic sampling
Computational Demand	15.5 days on 40-core processor for MCES computations	Highlights method's computational intensity
Outlier Identification	Certain lipid classes formed outlier clusters	Some compound classes dominate embeddings disproportionately
Distance Distribution	Most distances large, but minimum distances to neighbors usually <10	Sparse coverage with localized clusters

Methodologies for Assessing Chemical Space Coverage

Structural Distance Measurement Using MCES

The Maximum Common Edge Subgraph (MCES) distance provides a chemically meaningful measure of structural similarity that outperforms simpler fingerprint-based methods [55]. The MCES approach identifies the largest substructure common to two molecules, providing an alignment that captures chemical intuition better than traditional fingerprint methods [55].

Protocol: Myopic MCES Distance (mMCES) Calculation

• Problem Formulation: Represent molecules as graphs with atoms as nodes and bonds as edges
• Lower Bound Estimation: Compute provably correct lower bounds of all distances to filter trivial cases
• Exact Computation: Perform exact MCES computation only for distances below a set threshold (typically 10)
• Distance Assignment: Use exact distance if below threshold, otherwise use lower bound or threshold value
• Efficiency Optimization: Combine Integer Linear Programming with heuristic bounds to manage computational complexity

This method enables practical analysis of large datasets by reducing computational burden while preserving accuracy for chemically similar structures [55].

Dimensionality Reduction for Chemical Space Visualization

Dimensionality reduction (DR) techniques serve as essential tools for visualizing and assessing chemical space coverage through "chemography"—the creation of chemical space maps [57]. These techniques transform high-dimensional molecular descriptor data into human-interpretable 2D or 3D visualizations.

Table 2: Comparison of Dimensionality Reduction Methods for Chemical Space Analysis

Method	Type	Strengths	Weaknesses	Optimal Use Cases
PCA	Linear	Computational efficiency, reproducibility	Poor preservation of non-linear relationships	Initial data exploration, linearly separable data
t-SNE	Non-linear	Excellent neighborhood preservation	Computational intensity, perplexity sensitivity	Highlighting cluster separation in similar compounds
UMAP	Non-linear	Balance of local/global structure, faster than t-SNE	Parameter sensitivity, potential false connections	General-purpose chemical mapping with large datasets
GTM	Non-linear	Probabilistic framework, uncertainty quantification	Complex implementation, computational demand	Generating interpretable property landscapes

Protocol: Neighborhood Preservation Analysis

• Descriptor Calculation: Compute molecular representations (Morgan fingerprints, MACCS keys, ChemDist embeddings)
• Hyperparameter Optimization: Conduct grid-based search using percentage of preserved nearest neighbors as optimization metric
• Neighbor Definition: Define neighbors in both descriptor space (using Euclidean distance or 1-Tanimoto similarity) and latent space (using Euclidean distance)
• Metric Calculation: Compute neighborhood preservation metrics including:
  • PNN(k): Average number of preserved nearest neighbors
  • QNN(k): Co-k-nearest neighbor size
  • AUC(QNN): Area under QNN curve
  • LCMC: Local continuity meta criterion
  • Trustworthiness and Continuity
• Visual Assessment: Apply scatterplot diagnostics (scagnostics) to quantitatively assess visualization characteristics relevant to human perception [57]

Workflow for Coverage Assessment

The following diagram illustrates the comprehensive workflow for assessing chemical space coverage in molecular datasets:

Diagram 1: Chemical space coverage assessment workflow (76 characters)

Consequences for Potential Energy Surface Exploration

Impact on MLIP Development and Robustness

In the context of exploring potential-energy surfaces, coverage bias directly impacts the robustness and transferability of machine-learned interatomic potentials (MLIPs) [2]. The autoplex framework and similar automated approaches for MLIP development rely on comprehensive sampling of configurational space, including both local minima and highly unfavorable regions of the PES [2]. When training data lacks diversity, MLIPs may fail to accurately model rare events, transition states, or underrepresented molecular configurations, leading to potentially catastrophic failures in molecular dynamics simulations.

The consequences manifest particularly in binary systems with multiple phases of varied stoichiometric compositions [2]. For example, a model trained only on TiO2 may capture rutile and anatase polymorphs accurately but produces unacceptable errors (>1 eV at.⁻¹) when applied to rocksalt-type TiO or other stoichiometries [2]. This highlights the critical importance of comprehensive stoichiometric representation during training data construction.

Special Challenges in Low-Data Regimes

Molecular property prediction often operates in ultra-low data regimes, where the scarcity of reliable, high-quality labels impedes development of robust predictors [56]. Techniques like multi-task learning (MTL) aim to alleviate data bottlenecks by exploiting correlations among related molecular properties, but imbalanced training datasets often degrade efficacy through negative transfer [56].

Protocol: Adaptive Checkpointing with Specialization (ACS)

• Architecture Design: Implement shared task-agnostic backbone (GNN) with task-specific MLP heads
• Training Monitoring: Track validation loss for each task independently
• Checkpointing: Save best backbone-head pair when task validation loss reaches new minimum
• Specialization: Obtain task-specific model combining shared knowledge with specialized capability
• Negative Transfer Mitigation: Protect individual tasks from deleterious parameter updates while promoting beneficial inductive transfer

This approach has demonstrated capability to learn accurate models with as few as 29 labeled samples, dramatically reducing data requirements while maintaining prediction reliability [56].

Solutions and Best Practices

Strategic Dataset Construction and Curation

The creation of purpose-built quantum chemical databases aligned with industrial demands represents a crucial step toward addressing coverage bias [58]. Recent efforts have produced databases like ThermoG3 (53,550 structures), ThermoCBS (52,837 compounds), ReagLib20 (45,478 molecules), and DrugLib36 (40,080 compounds) specifically designed to cover diverse chemical spaces relevant to industrial applications [58]. These databases consider criteria including molecule size, heteroatom presence, and constituent elements to ensure broader coverage than traditional benchmarks like QM9.

Protocol: Representative Dataset Construction

• Domain Definition: Identify target chemical space based on application requirements (pharmaceuticals, energy materials, etc.)
• Diversity Metrics: Define diversity targets based on compound classes, elemental composition, and structural features
• Strategic Sampling: Implement maximum dissimilarity selection or cluster-based sampling to maximize coverage
• Bias Assessment: Apply MCES-based coverage analysis to identify underrepresented regions
• Iterative Expansion: Use active learning to strategically fill coverage gaps

The Scientist's Toolkit: Essential Research Reagents

Table 3: Essential Computational Tools for Chemical Space Analysis

Tool/Resource	Type	Function	Application Context
autoplex	Software Framework	Automated exploration and fitting of potential-energy surfaces	MLIP development for materials modelling [2]
MCES Distance	Algorithm	Structural similarity measurement based on maximum common edge subgraph	Chemical space coverage assessment and bias detection [55]
UMAP	Dimensionality Reduction	Non-linear projection for high-dimensional data visualization	Chemical space mapping and cluster identification [55] [57]
ACS Training	ML Method	Adaptive checkpointing with specialization for multi-task learning	Molecular property prediction in low-data regimes [56]
D-MPNN	Neural Architecture	Directed message-passing neural networks for molecular graphs	Molecular property prediction with 2D/3D structural information [58]
ClassyFire	Classification	Automated chemical compound classification	Compound class distribution analysis [55]
UR-1505	UR-1505, CAS:651331-92-9, MF:C10H7F5O4, MW:286.15 g/mol	Chemical Reagent	Bench Chemicals

Active Learning and Automated Exploration Frameworks

Automated frameworks like autoplex implement active learning strategies to iteratively optimize datasets by identifying rare events and selecting the most relevant configurations via suitable error estimates [2]. These approaches combine random structure searching (RSS) with MLIP fitting to explore configurational space efficiently without relying exclusively on costly ab initio molecular dynamics computations [2].

The following diagram illustrates the automated exploration and learning workflow for potential-energy surfaces:

Diagram 2: Automated PES exploration workflow (76 characters)

Protocol: Automated Potential-Energy Surface Exploration

• Initialization: Set up random structure searching (RSS) parameters and initial MLIP model
• Structure Generation: Generate diverse molecular configurations via RSS
• Candidate Evaluation: Evaluate configurations using current MLIP to identify promising candidates
• Uncertainty Quantification: Select structures with high prediction uncertainty for ab initio calculation
• Data Augmentation: Perform DFT single-point calculations (typically 100 per iteration) to augment training data
• Model Retraining: Update MLIP with augmented dataset
• Convergence Check: Evaluate model performance across target configurations; repeat until target accuracy (e.g., 0.01 eV at.⁻¹) is achieved

This approach has demonstrated robust performance across diverse systems including elemental silicon, TiOâ‚‚ polymorphs, and complex binary titanium-oxygen systems [2].

Ensuring model generalization requires confronting the fundamental challenges of dimensionality bias and chemical space coverage in molecular machine learning. The pitfalls are significant: models trained on non-uniform data fail to accurately predict properties for structures outside their narrow training domain, limiting their utility in real-world applications like drug discovery and materials design [55]. By implementing comprehensive coverage assessment using MCES-based distance metrics and strategic dimensionality reduction, researchers can identify and quantify these gaps [55] [57].

The integration of automated exploration frameworks like autoplex with active learning strategies represents a promising path forward [2]. These approaches enable efficient sampling of potential energy surfaces while strategically addressing coverage gaps through uncertainty-driven data acquisition. Combined with specialized training schemes like ACS for low-data regimes [56] and purpose-built databases targeting specific application domains [58], the field moves closer to achieving truly generalizable molecular models that maintain predictive accuracy across diverse chemical spaces.

As molecular machine learning continues to advance, maintaining rigorous attention to chemical space coverage will be essential for developing models that not only perform well on benchmark datasets but also deliver reliable predictions in the exploration of novel molecular structures and materials.

The exploration of potential energy surfaces (PES) is fundamental to predicting material properties, reaction mechanisms, and dynamical processes in computational chemistry and materials science. Machine learning (ML) has revolutionized this field by enabling large-scale atomistic simulations with quantum-mechanical accuracy through machine-learned interatomic potentials (MLIPs) [2]. However, a significant challenge persists: inconsistencies in reference data generated by different ab initio methods, functional choices, or computational parameters. These discrepancies propagate into the training phase of ML models, compromising their predictive reliability for properties such as defect energies, diffusion barriers, and phase stability [59].

This technical guide addresses the critical need for robust protocols to manage and mitigate these inconsistencies. We frame the discussion within the broader thesis of exploring PES with machine learning, providing researchers and drug development professionals with methodologies to enhance the consistency, accuracy, and reliability of their ML-driven simulations.

The Core Challenge: Discrepancies in Ab Initio Data

Discrepancies in ab initio data arise from various sources, including the choice of exchange-correlation functionals, basis sets, dispersion corrections, and treatment of electron correlation. When developing MLIPs, these inconsistencies manifest as errors in simulated properties, even when conventional metrics like root-mean-square error (RMSE) on energies and forces appear excellent.

Recent studies demonstrate that MLIPs with low average errors can still exhibit significant inaccuracies in reproducing atomistic dynamics and related properties. For example, MLIPs for silicon showed force RMSEs below 0.3 eV Å⁻¹ yet failed to accurately capture interstitial diffusion barriers [59]. Similarly, in the titanium-oxygen system, a model trained solely on TiO₂ data produced errors exceeding 1 eV atom⁻¹ when applied to rocksalt-type TiO, highlighting the transferability issues arising from inconsistent training data across stoichiometries [2].

Table 1: Common Sources of Discrepancies in Ab Initio Data for ML-PES

Source of Discrepancy	Impact on PES	Effect on MLIP Performance
Functional Choice (e.g., LDA vs. GGA vs. hybrid)	Systematic shifts in equilibrium geometries, reaction barriers, and binding energies	Biased prediction of phase stability and activation energies
Basis Set Completeness	Incomplete description of electron density, especially in anisotropic or weakly-bonded systems	Inaccuracies in simulating defect formation and molecular adsorption
Treatment of Dispersion Forces	Varying description of long-range interactions affecting layered materials and molecular crystals	Errors in predicting stacking energies and supramolecular assembly
k-point Sampling	Different numerical convergence for periodic systems, especially for metals and semiconductors	Artifacts in simulated phonon spectra and elastic constants

Methodological Frameworks for Consistency Management

Δ-Machine Learning (Δ-ML) Approach

The Δ-ML approach provides a powerful framework for managing discrepancies between different levels of theory. This method uses machine learning to correct a low-level ab initio PES towards a high-level reference, rather than learning the entire PES from scratch [4].

In practice, a flexible analytical PES or semi-empirical potential serves as the baseline. ML then learns the difference (Δ) between this baseline and high-level ab initio data. This strategy was successfully applied to the H + CH₄ hydrogen abstraction reaction, where a permutation invariant polynomial neural network (PIP-NN) surface corrected a lower-level analytical PES [4]. The resulting Δ-ML PES accurately reproduced both kinetics and dynamics information from the high-level surface, including variational transition state theory and quasiclassical trajectory results for the H + CD₄ reaction.

The experimental protocol for implementing Δ-ML involves:

• Generate Low-Level Reference Data: Perform extensive sampling of the configurational space using the low-level method (e.g., DFT with GGA functional)
• Acquire High-Level Correction Data: Compute high-level (e.g., CCSD(T)) single-point energies for strategically selected configurations
• Train Δ-Model: Learn the difference between high-level and low-level energies using appropriate descriptors
• Validation: Perform rigorous validation on independent test sets and compute target properties (e.g., reaction rates, diffusion coefficients)

Automated Active Learning Frameworks

Automated frameworks like autoplex address data consistency through iterative exploration and fitting of PES [2] [60]. These systems employ active learning to strategically expand training data into regions of configurational space where model uncertainty is high, ensuring consistent description across diverse atomic environments.

The autoplex framework implements random structure searching (RSS) combined with MLIP fitting, requiring only DFT single-point evaluations rather than costly ab initio molecular dynamics [2]. This approach automatically explores local minima and unfavorable regions of PES that must be included for robust potential development. For the Ti-O system, this method achieved accuracies of ~0.01 eV atom⁻¹ across multiple stoichiometries (Tiâ‚‚O₃, TiO, Tiâ‚‚O) by systematically expanding training data through thousands of automated iterations [2].

Table 2: Performance of Automated MLIP Frameworks Across Material Systems

Material System	Exploration Method	Structures Evaluated	Final Accuracy (RMSE)
Elemental Silicon	GAP-RSS	~500 for diamond/β-tin; ~5000 for oS24	~0.01 eV atom⁻¹
TiO₂ Polymorphs	GAP-RSS	Few thousand	<0.01 eV atom⁻¹ for rutile/anatase
Binary Ti-O System	GAP-RSS	Up to 5000	~0.01 eV atom⁻¹ for multiple stoichiometries
Crystalline/Liquid Water	GAP-RSS	Not specified	Quantum-mechanical accuracy for phases

Advanced Error Evaluation Metrics

Conventional MLIP assessment relying on average energy and force errors is insufficient for detecting discrepancies in dynamical properties. Novel evaluation metrics specifically targeting rare events and atomic dynamics provide more meaningful consistency measures [59].

Research shows that force errors on migrating atoms during rare events (e.g., vacancy or interstitial diffusion) serve as better indicators of MLIP performance for dynamical properties. By developing specialized testing sets like "interstitial-RE" and "vacancy-RE" configurations, researchers can quantify force errors specifically for atoms involved in diffusion processes [59]. MLIPs optimized using these targeted metrics demonstrate improved prediction of diffusion coefficients and energy barriers compared to those selected solely based on low average errors.

The protocol for implementing advanced error evaluation:

• Identify Critical Configurations: Extract snapshots from ab initio MD simulations containing migrating defects or transition states
• Create Specialized Testing Sets: Curate configurations representing rare events not adequately sampled in standard training
• Quantify Targeted Errors: Compute force errors specifically for atoms involved in the rare events
• Iterative Refinement: Use these metrics to guide active learning and hyperparameter optimization

Experimental Protocols and Workflows

Integrated Software Toolkits

Comprehensive software packages provide structured workflows for managing data consistency in ML-PES development. The Asparagus package offers a unified solution combining initial data sampling, ab initio calculations, ML model training, and evaluation [61]. Its modular architecture encompasses the entire ML-PES construction pipeline, ensuring reproducibility and reducing the initial hurdle for new users.

Similarly, autoplex builds on existing computational infrastructure, particularly the atomate2 framework underlying the Materials Project, ensuring interoperability with high-throughput computational materials science [2] [60]. These integrated toolkits implement best practices for data consistency by design, providing default parameters that yield reliable results while allowing expert customization.

Workflow for Multi-fidelity Data Integration

Managing discrepancies across different ab initio methods requires careful workflow design. The following protocol enables consistent MLIP development:

• Initial Exploration with Low-Level Method:

  • Perform random structure searching using low-level DFT (e.g., GGA functional)
  • Include diverse stoichiometries, phases, and defect configurations
  • Use automated frameworks like autoplex to guide exploration

• Strategic High-Level Corrections:

  • Identify critical configurations (transition states, weakly-bound complexes)
  • Compute single-point energies with high-level method (e.g., hybrid functional, CCSD(T))
  • Implement Δ-ML to correct the baseline PES

• Validation Across Properties:

  • Test on phonon spectra, elastic constants, and phase stability
  • Validate against experimental data where available
  • Compute dynamic properties (diffusion coefficients, reaction rates)

• Iterative Refinement:

  • Use active learning to identify regions of high uncertainty
  • Expand training data strategically, not exhaustively
  • Employ advanced error metrics to guide refinement

The Scientist's Toolkit: Essential Research Reagents

Table 3: Essential Software Tools for Managing Ab Initio Discrepancies in ML-PES

Tool/Resource	Function	Application Context
autoplex	Automated framework for PES exploration and MLIP fitting	High-throughput random structure searching across compositions
Asparagus	Modular workflow for autonomous ML-PES construction	User-guided PES development with reproducible methodologies
Gaussian Approximation Potential (GAP)	MLIP framework using SOAP descriptors	Data-efficient potential fitting compatible with active learning
OpenSPGen	Open-source tool for generating sigma profiles	Creating physically meaningful molecular descriptors for ML
Δ-ML Implementation	Correcting low-level PES with high-level data	Bridging accuracy-cost tradeoff in quantum chemistry methods

Addressing discrepancies from different ab initio methods requires a multifaceted approach combining Δ-ML methodologies, automated active learning frameworks, and advanced error metrics. By implementing the protocols and toolkits outlined in this guide, researchers can develop more consistent and reliable ML potentials for exploring potential energy surfaces. These strategies enable the community to move beyond simple error metrics toward robust validation of dynamical properties and rare events, ultimately enhancing the predictive power of machine learning in computational materials science and drug development.

The future of consistent ML-PES development lies in the intelligent integration of multi-fidelity data, where expensive high-level calculations are strategically deployed to correct systematic errors in more affordable methods, creating a virtuous cycle of improved accuracy and efficiency in computational materials discovery.

The exploration of potential energy surfaces (PES) is fundamental to advancements in materials science and drug development, dictating properties from catalytic activity to molecular stability. For decades, computational methods have navigated a persistent trade-off: achieving high prediction accuracy requires computationally expensive quantum mechanical methods like density functional theory (DFT), while faster, classical force fields often sacrifice quantum accuracy and reactivity. Machine-learned interatomic potentials (MLIPs) have emerged as a transformative solution, promising to bridge this divide. However, the development and deployment of MLIPs introduce their own performance optimization landscape, where strategic decisions directly influence the balance between computational cost and predictive fidelity. This guide details the methodologies and frameworks that enable researchers to construct efficient, accurate MLIPs for reliable PES exploration.

The core challenge lies in the fact that MLIPs are data-driven models; their accuracy is intrinsically linked to the quality and quantity of their training data, which is itself generated through costly ab initio calculations. Therefore, optimizing for performance is not a single-step process but an integrated strategy encompassing data generation, model architecture selection, and targeted sampling. The recent advent of large-scale, community-driven datasets and automated training frameworks has begun to reshape this landscape, offering new pathways to robust models without prohibitive computational investment.

Foundational Concepts and Quantitative Benchmarks

Performance Metrics for MLIPs

The performance of an MLIP is quantified along two primary axes: its prediction accuracy and its computational cost. Accuracy is typically measured against a reference method (e.g., DFT) using metrics like Root Mean Square Error (RMSE) or Mean Absolute Error (MAE) for energies and forces. Computational cost encompasses the expenses of dataset generation (DFT calculations) and the model's inference speed during simulation.

A key development is the emergence of foundational models and large-scale datasets that establish new performance baselines. For instance, models trained on Meta's Open Molecules 2025 (OMol25) dataset—containing over 100 million molecular snapshots—demonstrate that extensive, chemically diverse data is crucial for high accuracy across a broad range of systems [62] [49]. The computational cost of creating such a dataset was monumental, requiring over six billion CPU hours, but the resulting pre-trained models offer a high-accuracy starting point that drastically reduces the need for new, system-specific DFT calculations [49].

Table 1: Comparative Analysis of Machine Learning Potentials for Molecular Systems.

Model/Dataset	Key Architectural Feature	Reported Energy MAE	Reported Force MAE	Computational Cost Factor
EMFF-2025 [63]	Neural Network Potential (NNP)	< 0.1 eV/atom	< 2 eV/Ã…	Lower than DFT; uses transfer learning
OMol25 eSEN [62]	Equivariant Transformer	Matches DFT on benchmarks	N/A	Pre-trained; inference is ~10,000x faster than DFT
Δ-ML (H + CH₄ PES) [4]	Corrects low-level PES with high-level data	Reproduces high-level kinetics/dynamics	N/A	Highly cost-effective vs. full high-level calculation
GAP-RSS (autoplex) [2]	Gaussian Approximation Potential	~0.01 eV/atom (for Si)	N/A	Automated sampling reduces required DFT calculations

Table 2: Accuracy and Cost of Dataset Generation Methods.

Data Generation Method	Description	Relative Computational Cost	Best For
Active Learning [2]	Iteratively samples configurations based on model uncertainty	Medium	Exploring complex reactions and rare events
Random Structure Searching (RSS) [2]	Randomly generates structures to explore configurational space	Medium to High	Discovering unknown stable/metastable phases
Ab Initio Molecular Dynamics (AIMD)	Samples configurations from dynamics trajectories	High	Sampling thermodynamic ensembles
Leveraging Foundational Datasets (e.g., OMol25) [62] [49]	Fine-tuning pre-trained models on limited custom data	Low	Rapid application to new systems within covered chemical space

Experimental Protocols for Efficient MLIP Development

Protocol 1: The autoplex Framework for Automated PES Exploration

The autoplex framework automates the iterative process of exploring a PES and fitting an MLIP, significantly reducing manual effort and optimizing the use of computational resources [2].

Detailed Methodology:

• Initialization: Define the chemical system (elements, composition ranges) and select a prior potential (which can be very simple).
• Random Structure Generation: The framework automatically generates a large number of random initial atomic structures.
• Structure Relaxation with MLIP: These structures are relaxed using the current iteration of the MLIP (e.g., a Gaussian Approximation Potential, GAP), not DFT. This is a computationally cheap step.
• DFT Single-Point Calculations: A subset of the relaxed structures is selected for single-point energy and force calculations using DFT. This selection can be based on criteria like structural diversity or energy.
• Model Training and Refinement: The results from the DFT calculations are added to the training dataset, and the MLIP is retrained.
• Iteration: Steps 2-5 are repeated, with each new iteration using an improved MLIP to drive the search. The loop continues until the prediction error for key structures of interest falls below a predefined threshold (e.g., 0.01 eV/atom) [2].

This protocol is highly efficient because it minimizes the number of expensive DFT calculations—using them only for single-point evaluations on MLIP-prescreened structures—and fully automates the workflow. It has been validated on systems ranging from elemental silicon to the complex binary Ti-O system [2].

Protocol 2: Transfer Learning with Pre-Trained Foundational Models

For many applications, fine-tuning a large, pre-trained model is more efficient than training a model from scratch. This protocol leverages models trained on massive datasets like OMol25.

Detailed Methodology:

• Model Selection: Choose a suitable pre-trained model that covers the chemical elements of your system. Examples include the Universal Models for Atoms (UMA) or eSEN models released by Meta's FAIR team [62].
• Target Data Generation: Perform a limited set of ab initio calculations (e.g., DFT) specifically for your system of interest. This dataset must capture the relevant configurations but can be orders of magnitude smaller than what is needed for training from scratch.
• Fine-Tuning: Continue training the pre-trained model on your new, smaller dataset. This process "steers" the general-purpose model towards the specific physics and chemistry of your target system.
• Validation: Rigorously validate the fine-tuned model on a held-out test set of your target data, ensuring it maintains the accuracy of the foundational model while now being specialized for your application.

The EMFF-2025 potential for energetic materials is a prime example, developed using a transfer learning scheme from a pre-trained model, which allowed it to achieve DFT-level accuracy for 20 high-energy materials with minimal new data from DFT calculations [63].

Visualizing Workflows and Logical Relationships

ML Model Fine-Tuning Workflow

Automated PES Exploration Loop

The Scientist's Toolkit: Essential Research Reagent Solutions

Table 3: Key Software Tools and Datasets for MLIP Development.

Tool / Resource	Type	Primary Function	Reference / Source
OMol25 Dataset	Dataset	Massive, diverse training set of 100M+ molecular snapshots for robust model training.	[62] [49]
autoplex	Software Framework	Automated workflow for exploring PES and fitting MLIPs from scratch.	[2]
Deep Potential (DP)	MLIP Architecture	A scalable NNP framework for complex reactive processes and large-scale systems.	[63]
Gaussian Approximation Potential (GAP)	MLIP Architecture	Data-efficient MLIP method, often used with RSS for initial PES exploration.	[2]
eSEN & UMA Models	Pre-trained MLIPs	Foundational models offering high accuracy out-of-the-box, suitable for fine-tuning.	[62]
Δ-Machine Learning (Δ-ML)	Methodology	Corrects inexpensive PES with high-level data for cost-effective accuracy.	[4]

Optimizing the balance between computational cost and prediction accuracy is a dynamic and multi-faceted endeavor. The field is rapidly moving away from building isolated, hand-crafted models and towards a paradigm of leveraging foundational datasets and automated frameworks. As summarized in this guide, strategies such as automated active learning with autoplex and transfer learning from pre-trained models like UMA provide concrete, actionable pathways for researchers to achieve high-accuracy simulations of potential energy surfaces at a fraction of the traditional cost. The ongoing development of even larger and more chemically diverse datasets, coupled with more efficient model architectures and training techniques, promises to further dissolve the trade-off, accelerating discovery in materials science and drug development.

Best Practices for Robust and Reproducible ML-PES Development

The development of Machine-Learned Potential Energy Surfaces (ML-PES) has revolutionized atomistic simulations, enabling large-scale molecular dynamics with quantum-mechanical accuracy. This paradigm shift is fundamental to advancements in materials modelling, drug discovery, and computational chemistry [2]. However, the transition from hand-crafted, domain-specific models to robust, general-purpose potentials introduces significant challenges in ensuring their reliability and reproducibility across diverse chemical spaces. This guide synthesizes current methodologies and practical recommendations for constructing ML-PES that are both chemically accurate and reproducible, framing them within the broader thesis of exploring potential-energy surfaces with machine learning research. We focus on the end-to-end pipeline, from initial data generation to final model validation, providing a structured approach for researchers and drug development professionals.

Theoretical Foundations of ML-PES

A Potential Energy Surface (PES) represents the energy of a system as a function of the positions of its constituent atoms. It is the cornerstone for understanding molecular structure, reactivity, and dynamics. The primary objective of an ML-PES is to learn this multidimensional hypersurface from a finite set of high-level ab initio calculations, thereby creating a surrogate model that provides accurate energies and forces at a fraction of the computational cost.

Several machine learning architectures have been successfully applied to this task, each with distinct advantages:

• Gaussian Approximation Potentials (GAP): Utilize Gaussian process regression to provide not only predictions but also inherent uncertainty quantification. This feature is particularly valuable for guiding active learning strategies [2].
• Neural Network Potentials (NNPs): Employ various neural network architectures to capture complex, non-linear relationships in atomic configurations. A prominent subtype is the Permutationally Invariant Polynomial Neural Network (PIP-NN), which builds in physical symmetries such as permutation invariance of like atoms [1].
• Δ-Machine Learning (Δ-ML): A highly efficient strategy that corrects a cheap, low-level (LL) potential, such as one from an analytical functional form or Density-Functional Tight-Binding (DFTB), with a machine-learned correction term trained on a small set of high-level (HL) data. The core equation is expressed as: ( V^{HL}(\mathbf{R}) = V^{LL}(\mathbf{R}) + \Delta V^{HL-LL}(\mathbf{R}) ) where ( \mathbf{R} ) represents the atomic coordinates, ( V^{LL} ) is the energy from the low-level method, and ( \Delta V^{HL-LL} ) is the machine-learned correction [1]. This approach can dramatically reduce the number of costly HL calculations required.

The ML-PES Development Workflow: A Roadmap to Robustness

A reproducible ML-PES development process is built on a structured, iterative workflow that emphasizes automation and systematic validation. The following diagram outlines the core stages.

Automated Configurational Space Exploration

The robustness of an ML-PES is fundamentally constrained by the diversity and quality of its training data. Manually generating datasets is a major bottleneck and can introduce biases. Automated exploration is therefore critical.

• Random Structure Searching (RSS): Methods like Ab Initio Random Structure Searching (AIRSS) generate a wide array of initial atomic configurations, including high-energy states and transition pathways, which are crucial for teaching the potential about the entire PES, not just local minima [2].
• Interoperable Workflow Automation: Frameworks like autoplex demonstrate the power of automation by integrating RSS with MLIP fitting and high-performance computing. This allows for high-throughput sampling and iterative model improvement without manual intervention, directly enhancing reproducibility [2]. The initial exploration should be agnostic to the final application to ensure broad coverage of the configurational space.

Table 1: Target Accuracies for ML-PES in Different Applications

Application Domain	Target Energy Accuracy (per atom)	Key Configurations to Sample
Static Material Properties	~0.01 eV/atom (≈ 0.1 eV/atom for phases) [2]	Crystalline polymorphs, defect structures, surfaces
Reaction Kinetics	< 1 kcal/mol (≈ 0.04 eV/atom) [1]	Transition states, minimum energy paths, reactant/product basins
Molecular Dynamics	~0.01 eV/atom for stability [2]	Liquid phases, amorphous systems, high-temperature configurations

Iterative Model Fitting and Active Learning

A single round of training on an initial dataset is often insufficient. An iterative, closed-loop process is a best practice for achieving robustness.

• Uncertainty-Guided Sampling: The model's own uncertainty estimates, inherent in methods like GAP, are used to identify regions of the configurational space where its predictions are unreliable. New ab initio calculations are then targeted at these uncertain configurations, efficiently improving the model with minimal data [2].
• Performance-Error Monitoring: The model's error on a held-out test set of known, relevant structures (e.g., key crystalline polymorphs or reaction barriers) should be tracked across iterations. As shown in benchmarks, the root mean square error (RMSE) for structures like the oS24 silicon allotrope or TiO2-B polymorph decreases systematically with an increasing number of strategically chosen single-point evaluations [2].

Comprehensive Model Validation and Benchmarking

Validation is the cornerstone of reproducibility. An ML-PES must be tested on properties it was not directly trained against.

• Static Property Prediction: Evaluate the model's accuracy on the energies and forces of known stable and metastable structures not included in the training set.
• Dynamic and Thermodynamic Properties: Conduct molecular dynamics simulations to compute properties like radial distribution functions, diffusion coefficients, or phase transition temperatures. Compare these results with experimental data or direct ab initio MD simulations where possible.
• Reaction Profile Accuracy: For studies involving chemical reactivity, the model must correctly reproduce the energy profile of key reactions, including reaction energies and barrier heights, to within chemical accuracy (1 kcal/mol) [1].

Table 2: Validation Protocols for ML-PES

Validation Type	Key Metrics	Reference Method
Energetics & Geometry	Formation energies, forces, vibrational frequencies	High-level ab initio (e.g., CCSD(T), DFT with validated functional)
Molecular Dynamics	Radial distribution functions, density, thermal expansion	Experimental data or ab initio MD
Kinetics & Reactivity	Reaction barrier heights, reaction energies, transition state geometries	High-level quantum chemistry calculations or experimental kinetics

The Scientist's Toolkit: Essential Research Reagents

Building a robust ML-PES relies on a suite of software tools and data resources. The following table details the key "research reagents" essential for modern development.

Table 3: Essential Research Reagents for ML-PES Development

Tool / Resource	Type	Primary Function	Application Note
autoplex [2]	Software Framework	Automated workflow for exploration and fitting of PES	Enables high-throughput, reproducible MLIP generation; interoperable with atomate2.
GAP (Gaussian Approximation Potential) [2]	MLIP Architecture	Fitting PES using Gaussian process regression	Valued for data efficiency and native uncertainty quantification.
PIP-NN (Permutationally Invariant Polynomial-Neural Network) [1]	MLIP Architecture	Constructing PES with built-in permutation invariance	Highly accurate for molecular systems (e.g., H + CH4 reaction).
Δ-ML (Delta-Machine Learning) [1]	Methodology	Correcting low-level PES with ML for high-level accuracy	Reduces computational cost; can use analytical PES or DFTB as low-level method.
AIRSS (Ab Initio Random Structure Searching) [2]	Methodology	Exploring configurational space to generate diverse training data	Critical for finding rare events and building comprehensive datasets.

Case Study: The H + CH4 Reaction and the Ti-O System

Δ-ML for a Polyatomic Reaction

The hydrogen abstraction reaction, H + CH4 → H2 + CH3, serves as a benchmark for polyatomic PES development. A recent study demonstrated the application of Δ-ML, using an analytical valence-bond/molecular mechanics (VB-MM) potential as the low-level (LL) reference and a high-accuracy PIP-NN surface as the high-level (HL) target [1]. The resulting Δ-ML PES successfully reproduced kinetics and dynamics information from the high-level surface, achieving near-chemical accuracy for the reaction barrier height (∼14.7 kcal/mol) with significantly reduced computational effort. This validates Δ-ML as a powerful strategy for complex, polyatomic systems.

Automated Exploration of a Binary Materials System

The development of a potential for the titanium-oxygen (Ti-O) system highlights the importance of stoichiometric diversity in training data. When an ML-PES was trained solely on TiO2 polymorphs (e.g., rutile, anatase), it failed catastrophically for other compositions like rocksalt-type TiO, with errors exceeding 1 eV/atom [2]. In contrast, using an automated framework like autoplex to explore the full binary Ti-O space yielded a single, robust potential that accurately described multiple phases with different stoichiometries (Ti2O3, TiO, Ti2O). This underscores that automation is not just an efficiency gain but a necessity for creating transferable and reliable models for complex materials systems.

The development of robust and reproducible ML-PES is maturing from a specialized craft into a more standardized engineering discipline. This transition is driven by several key pillars: the automation of configurational sampling to eliminate human bias, the adoption of iterative active learning for data-efficient model improvement, the implementation of rigorous and multi-faceted validation protocols, and the strategic use of methods like Δ-ML to maximize accuracy per computational dollar. By adhering to these best practices and leveraging emerging open-source frameworks, researchers can construct reliable ML-PES that will accelerate the exploration of complex potential-energy surfaces, ultimately driving discovery in materials science and drug development.

Benchmarking ML-PES Performance: Validation Protocols and Model Comparisons for Confident Deployment

In machine learning interatomic potentials (MLIPs), robust validation is the cornerstone of reliable and transferable models for exploring potential energy surfaces (PES). Moving beyond simple energy and force errors to a multi-faceted validation strategy is critical for ensuring model fidelity across diverse chemical environments and physical properties. This technical guide outlines the core quantitative metrics, detailed experimental protocols, and advanced validation methodologies essential for developing MLIPs that can be trusted in high-stakes applications, such as drug development and materials discovery.

The exploration of potential energy surfaces (PES) with machine-learned interatomic potentials (MLIPs) has become a powerful paradigm in computational chemistry and materials science [2]. MLIPs enable large-scale atomistic simulations with quantum-mechanical accuracy, facilitating research ranging from protein folding to the design of novel catalytic materials. However, the sophistication of an MLIP's architecture is secondary to the quality of its validation. A model that performs well only on a narrow, "easy" subset of configurations is of little practical use for exploratory research. Therefore, establishing a comprehensive suite of validation metrics is paramount. This guide details a holistic framework for MLIP validation, extending from foundational energy and force errors to sophisticated tests of predictive performance on challenging, out-of-sample configurations.

Core Quantitative Metrics

The most immediate measures of an MLIP's performance are the errors in its predictions of energies and forces compared to reference quantum-mechanical calculations, typically from Density-Functional Theory (DFT). These metrics provide a quantitative baseline for model accuracy. The following table summarizes the key metrics and their interpretations.

Table 1: Core Quantitative Validation Metrics for MLIPs

Metric Name	Mathematical Formulation	Physical Interpretation	Target Performance
Energy RMSE	$\text{RMSE}E = \sqrt{ \frac{1}{N} \sum{i=1}^{N} (Ei^{\text{ML}} - Ei^{\text{DFT}})^2 }$	Overall accuracy of the potential energy surface shape.	< 10 meV/atom for chemical accuracy [2]
Force RMSE	$\text{RMSE}F = \sqrt{ \frac{1}{3N} \sum{i=1}^{N} \sum{\alpha=1}^{3} (F{i,\alpha}^{\text{ML}} - F_{i,\alpha}^{\text{DFT}})^2 }$	Accuracy of atomic forces, critical for MD stability.	~100 meV/Ã… (system-dependent)
Energy MAE	$\text{MAE}E = \frac{1}{N} \sum{i=1}^{N}	Ei^{\text{ML}} - Ei^{\text{DFT}}	$	Robust measure of central tendency for energy error.	Similar to RMSE targets
Force MAE	$\text{MAE}F = \frac{1}{3N} \sum{i=1}^{N} \sum_{\alpha=1}^{3}	F{i,\alpha}^{\text{ML}} - F{i,\alpha}^{\text{DFT}}	$	Robust measure of central tendency for force error.	Similar to RMSE targets

It is critical to note that reporting only the overall error on a dataset can be misleading. As with other deep learning models, performance can be skewed by "easy" test cases [64]. A robust validation protocol requires stratifying these errors based on the difficulty or nature of the atomic configuration, such as reporting separate errors for different crystal polymorphs or for regions of the PES sampled via different methods (e.g., random structure searching versus molecular dynamics) [2] [64].

Experimental Protocols for Validation

Iterative Model Training and Active Learning

A modern best practice for developing robust MLIPs is to use an automated, iterative framework that integrates model training with data generation. The following workflow, implementable through software packages like autoplex, exemplifies this protocol [2].

Workflow Title: Iterative MLIP Training & Validation

Protocol Details:

• Initialization: Begin with a small, diverse set of atomic configurations (e.g., different bulk phases, surfaces, defects) with pre-computed DFT energies and forces.
• Model Training: Train an initial MLIP (e.g., a Gaussian Approximation Potential (GAP) [2] or neural network potential) on this dataset.
• PES Exploration: Use the current MLIP to drive a Random Structure Search (RSS) for new, low-energy, or high-error configurations. This step is crucial for exploring unseen regions of the PES without the cost of DFT molecular dynamics [2].
• Candidate Selection: From the RSS results, select structures that are chemically sensible but for which the MLIP's prediction uncertainty is high.
• DFT Verification: Perform single-point DFT calculations on the selected candidate structures to obtain new reference data.
• Data Augmentation: Add the new structures and their DFT-calculated energies/forces to the training dataset.
• Validation Check: Evaluate the retrained model's performance on a held-out validation set containing known polymorphs and challenging configurations.
• Convergence: Iterate steps 2-7 until the error metrics on the validation set fall below a predefined threshold (e.g., energy RMSE < 10 meV/atom) [2].

Stratified Validation Set Design

To avoid the pitfall of "easy test sets" [64], the validation data must be carefully curated.

Protocol Details:

• Stratification: Partition the validation set into distinct challenge levels. For materials, this could be based on:
  • Easy: Common bulk crystal structures with high symmetry.
  • Moderate: Less common polymorphs or structures with lower symmetry.
  • Hard: Metastable phases, defect-rich structures, surfaces, or configurations with chemical environments far from those in the training set [2] [64].
• Performance Reporting: Report energy and force errors (RMSE, MAE) separately for each stratification level. This reveals whether the model has truly learned the underlying physics or is merely performing well on simple cases.
• Real-World Distribution: Where possible, model the proportion of easy, moderate, and hard problems based on the expected distribution in real-world applications (e.g., the proportion of "twilight zone" proteins in a newly sequenced genome) [64].

Advanced and Application-Specific Metrics

While energy and force errors are necessary, they are not sufficient. A truly validated model must reproduce key experimental or high-fidelity computational observables.

Table 2: Advanced Application-Specific Validation Metrics

Application Domain	Key Validation Metrics	Computational Protocol
Catalysis & Reaction Dynamics	Reaction rates, free energy barriers, kinetic isotope effects.	Calculate using the MLIP in Transition State Theory (TST) or quasiclassical trajectory calculations, comparing results to high-level quantum chemistry data [65].
Phase-Change Materials	Relative phase stability, transition pressures, melting temperature, radial distribution functions.	Perform molecular dynamics (MD) or Monte Carlo (MC) simulations to compute phase diagrams and structural properties.
Biomolecular Simulations	Protein-ligand binding affinities, conformational equilibrium, solvation free energies.	Run long-timescale MD simulations and use methods like free energy perturbation (FEP) or umbrella sampling.
Mechanical Properties	Elastic constants (C11, C12, C44), tensile strength, phonon dispersion spectra.	Perform deformation simulations on crystal structures and lattice dynamics calculations.

The Scientist's Toolkit: Essential Research Reagents

The following table details key software and methodological "reagents" essential for the experiments and validation protocols described in this guide.

Table 3: Essential Research Reagent Solutions for MLIP Development

Item Name	Function / Purpose	Relevant Context
autoplex	An open-source, automated framework for iterative exploration and fitting of potential-energy surfaces [2].	High-throughput workflow management for MLIP development, integrates RSS, DFT, and fitting.
Gaussian Approximation Potential (GAP)	A machine-learning interatomic potential framework based on kernel regression and SOAP descriptors, known for data efficiency [2].	The MLIP engine used within autoplex and other workflows for initial PES exploration.
Δ-Machine Learning (Δ-ML)	A method to correct a low-level PES using a small number of high-level calculations, improving accuracy cost-effectively [65].	Creating high-level PES for kinetics and dynamics studies without exhaustive high-level computation.
Random Structure Searching (RSS)	A computational method for discovering stable and metastable crystal structures by randomly generating and relaxing atomic configurations [2].	Core component of the iterative training workflow for expanding the training dataset into unexplored PES regions.
Stratified Validation Set	A curated dataset where configurations are categorized by their level of difficulty or similarity to the training data [64].	Critical for diagnosing model weaknesses and ensuring performance across easy, moderate, and hard test cases.

Establishing validation metrics for machine-learned interatomic potentials is a multi-dimensional challenge that extends far beyond the simplistic reporting of energy and force RMSE. A rigorous validation protocol must incorporate iterative model refinement through active learning, employ stratified validation sets to expose model weaknesses, and verify performance against application-specific properties. By adopting the comprehensive framework outlined in this guide—encompassing quantitative metrics, detailed experimental protocols, and advanced validation techniques—researchers can develop MLIPs with the robustness and reliability required to confidently explore the complex potential energy surfaces that underpin drug discovery and advanced materials design.

The development of universal machine learning interatomic potentials (MLIPs) promises to revolutionize atomistic simulations by replacing expensive quantum-mechanical calculations. However, their robustness across different structural dimensionalities—from zero-dimensional (0D) molecules and clusters to three-dimensional (3D) bulk solids—remains a critical frontier for their reliable application in exploring potential energy surfaces (PES). This technical guide synthesizes recent benchmarking studies that quantitatively assess the performance of state-of-the-art universal models across this dimensional spectrum. The findings reveal a pronounced performance gap, where models excel in 3D bulk systems but show progressively degraded accuracy in lower-dimensional structures. This whitepaper details the benchmarking methodologies, summarizes key quantitative results, and provides protocols for researchers to evaluate and apply these models in the context of drug development and materials science, with a specific focus on navigating complex PES.

The accurate and efficient computation of potential energy surfaces (PES) is a cornerstone for predicting reaction rates, spectroscopic properties, and dynamical processes in chemistry and materials science. Machine-learned interatomic potentials (MLIPs) have emerged as a powerful tool to overcome the prohibitive cost of high-level ab initio calculations, enabling large-scale molecular dynamics and crystal structure searches with near-quantum accuracy [2] [18]. A significant trend in the field is the development of "universal" or "foundational" models trained on extensive datasets, aiming to make accurate predictions for arbitrary atomic structures and compositions [66] [67].

A paramount challenge in this pursuit is the vast diversity of atomic environments found in nature, particularly when categorized by system dimensionality. These range from:

• 0D (Zero-Dimensional): Isolated molecules and atomic clusters.
• 1D (One-Dimensional): Nanowires, nanoribbons, and nanotubes.
• 2D (Two-Dimensional): Atomic layers, slabs, and surfaces.
• 3D (Three-Dimensional): Bulk crystals and amorphous solids.

The local atomic environments, coordination numbers, and electronic structures differ significantly across these categories. Consequently, a model that performs exceptionally well on bulk 3D crystals may fail when applied to a 2D surface or a 0D molecule. This guide synthesizes recent benchmarking efforts that systematically evaluate model performance across this dimensional spectrum, providing a crucial resource for researchers, particularly in drug development where molecular solids (0D/3D hybrids) and surface interactions are of paramount importance.

Quantitative Benchmarking: Performance Across Dimensionalities

A dedicated benchmark designed to evaluate the predictive capabilities of universal MLIPs across varying dimensionalities provides clear, quantitative evidence of this performance disparity [66]. The benchmark tested multiple state-of-the-art models on a suite of systems including molecules and clusters (0D), nanowires and nanotubes (1D), atomic layers and slabs (2D), and bulk materials (3D).

Table 1: Benchmarking Results for Universal Machine Learning Interatomic Potentials Across Different Dimensionalities [66]

Dimensionality	Example Systems	Best Performing Models	Average Error in Atomic Positions (Ã…)	Average Error in Energy (meV/atom)
0D (Molecules/Clusters)	Isolated molecules, atomic clusters	Orbital Version 2, EquiformerV2, Equivariant Smooth Energy Network	0.01 - 0.02	< 10
1D (Nanowires/Nanoribbons)	Nanowires, nanotubes, nanoribbons	Orbital Version 2, EquiformerV2, Equivariant Smooth Energy Network	0.01 - 0.02	< 10
2D (Atomic Layers/Slabs)	Atomic layers, slabs, surfaces	Orbital Version 2, EquiformerV2, Equivariant Smooth Energy Network	0.01 - 0.02	< 10
3D (Bulk Solids)	Bulk crystals, amorphous solids	Orbital Version 2, EquiformerV2, Equivariant Smooth Energy Network	0.01 - 0.02	< 10

The key finding is that while all tested models demonstrated excellent performance for three-dimensional systems, accuracy degraded progressively for lower-dimensional structures [66]. The best-performing models, however, managed to achieve errors in atomic positions in the range of 0.01-0.02 Ã… and errors in energy below 10 meV/atom on average across all dimensionalities. This demonstrates that state-of-the-art universal MLIPs have reached a level of accuracy that allows them to serve as direct replacements for Density Functional Theory (DFT) calculations for a wide range of simulations, albeit at a fraction of the computational cost [66].

Experimental and Methodological Protocols

The reliability of any benchmark is contingent upon rigorous and reproducible methodologies. This section outlines the core protocols employed in the cited studies for data generation, model training, and performance evaluation.

Data Generation and Dataset Design

A critical factor in training and benchmarking robust MLIPs is the quality and diversity of the underlying data. Traditional datasets often focus primarily on equilibrium structures, limiting their applicability for simulations that explore the full PES, including transition states and high-energy configurations [67].

• The MAD Dataset Philosophy: The Massive Atomic Diversity (MAD) dataset addresses this by being designed to encompass a broad spectrum of atomic configurations, including both organic and inorganic systems [67]. It is constructed by starting with stable structures and then aggressively applying systematic perturbations, such as rattling atoms and randomizing compositions, to achieve massive coverage of the configurational space. This ensures that models trained on MAD encounter not just low-energy minima but also the distorted configurations crucial for dynamics and phase transition studies.
• Consistent Computational Settings: The MAD dataset employs a consistent level of theory across all ab initio calculations to ensure a coherent structure-energy mapping [67]. This avoids errors introduced by mixing computational parameters, which is a common issue when aggregating data from multiple sources.

Automated Active Learning and Potential Fitting

The manual generation of training data is a major bottleneck in MLIP development. Automated frameworks like autoplex have been introduced to streamline the exploration and fitting of PES [2] [18].

• Iterative Exploration and Learning: The autoplex framework implements an automated iterative cycle. It starts with a small set of ab initio data, trains an initial MLIP, and then uses this potential to drive random structure searching (RSS) to explore new regions of the PES [2]. The most informative configurations from these searches (e.g., those with high predictive uncertainty) are then selected for subsequent ab initio single-point calculations and added to the training set. This active-learning loop continues until the model achieves a target accuracy across a set of known structures and phases.
• Software Interoperability: These frameworks are designed for interoperability with high-performance computing systems and widely-used MLIP architectures, such as Gaussian Approximation Potentials (GAP), enabling high-throughput and automated potential development [2].

The following workflow diagram illustrates this automated iterative process for exploring and learning potential-energy surfaces:

Benchmarking and Out-of-Distribution (OOD) Generalization

Systematic benchmarking is essential to expose the limitations and strengths of ML models. The BOOM (Benchmarking Out-Of-distribution Molecular property predictions) study highlights a critical challenge: ML models often struggle to generalize to data that is outside their training distribution [68].

• OOD Splitting Methodology: BOOM benchmarks OOD generalization by splitting datasets with respect to the target property values. A kernel density estimator is fitted to the property distribution, and the molecules with the lowest probabilities (the tail ends of the distribution) are assigned to the OOD test set [68]. This directly tests a model's ability to extrapolate to novel, high-performing molecules, which is the essence of molecular discovery.
• Key Finding: The BOOM benchmark, evaluating over 140 model-task combinations, found that no existing model achieved strong OOD generalization across all tasks. The top-performing model exhibited an average OOD error three times larger than its in-distribution error [68]. This underscores that high accuracy on a standard test set does not guarantee performance in discovery-oriented applications.

For researchers embarking on exploring PES with MLIPs, a suite of software tools and datasets has become indispensable. The table below catalogs key "research reagent solutions" referenced in this guide.

Table 2: Essential Computational Tools and Datasets for ML-Driven PES Exploration

Tool / Dataset	Type	Primary Function	Relevance to PES Exploration
Autoplex [2]	Software Framework	Automated exploration and fitting of MLIPs.	Automates the active-learning loop for robust potential generation, minimizing manual effort.
MAD Dataset [67]	Dataset	A compact, diverse set of atomic structures and properties.	Provides massive atomic diversity for training MLIPs that generalize to non-equilibrium structures.
Matbench [69]	Benchmark Suite	A test suite of 13 materials property prediction tasks.	Provides a standardized framework for evaluating and comparing the performance of different ML models.
Gaussian Approximation Potential (GAP) [2]	MLIP Framework	A method for fitting interatomic potentials using Gaussian process regression.	Often used for its data efficiency in active-learning and structure-search applications.
BOOM Benchmark [68]	Benchmark Suite	A benchmark for out-of-distribution molecular property prediction.	Evaluates a model's ability to extrapolate, which is crucial for genuine molecular discovery.

Implications for Drug Development and Materials Science

The ability to accurately simulate systems of mixed dimensionality has direct and profound implications for drug development professionals and materials scientists.

• Drug Polymorph Prediction: The determination of crystal structures of active pharmaceutical ingredients (APIs) is critical, as different polymorphs can have vastly different bioavailability and stability. Machine learning models have been successfully applied to predict NMR chemical shifts in molecular solids like cocaine and other drug compounds directly from crystal structures [70]. Accurate MLIPs are the foundation for reliably generating these structures through crystal structure prediction (CSP) protocols, which require navigating the complex PES of molecular packing.
• Simulating Interfaces and Complex Systems: The benchmarking results showing competent performance across dimensionalities indicate that the best models "already enable efficient simulations of complex systems containing subsystems of mixed dimensionality, opening new possibilities for modeling realistic materials and interfaces" [66]. This is essential for studying phenomena like drug adsorption on nanoparticle surfaces (0D-3D interface) or the interaction of APIs with biological membranes (3D-2D interface).

The following diagram maps the logical workflow for applying these ML tools to a real-world problem like drug polymorph prediction:

The benchmarking of universal machine learning models across dimensionalities reveals a field in rapid advancement. While a performance gap exists for lower-dimensional systems, state-of-the-art models have reached a significant milestone, achieving high accuracy from 0D molecules to 3D bulk materials. This progress, coupled with automated frameworks for PES exploration and carefully designed datasets, is paving the way for reliable, large-scale atomistic simulations in drug development and materials science. However, the challenge of robust out-of-distribution generalization remains a key frontier. For researchers, this underscores the importance of not only selecting high-performing universal models but also rigorously validating them against system-specific, out-of-distribution benchmarks relevant to their particular discovery goals. The continued development and application of these benchmarking and automation tools will be instrumental in realizing the full potential of machine learning to navigate the complex energy landscapes that govern molecular and materials behavior.

The accurate and efficient exploration of potential energy surfaces (PES) is a fundamental challenge in computational materials science and drug discovery. Machine Learning Interatomic Potentials (MLIPs) have emerged as transformative tools that bridge the gap between quantum-mechanical accuracy and classical molecular dynamics efficiency [25]. This whitepaper provides a comparative analysis of four leading universal MLIP architectures—MACE, Orbital (ORB), eSEN, and EquiformerV2—evaluating their performance, scalability, and applicability for PES exploration in scientific research and pharmaceutical development.

Foundational Principles

Modern MLIPs have evolved from using handcrafted invariant descriptors to sophisticated equivariant architectures that explicitly embed physical symmetries including translation, rotation, and reflection (E(3) equivariance) [25]. These advancements ensure that scalar predictions (e.g., total energy) remain invariant while vector and tensor targets (e.g., forces) exhibit correct equivariant behavior, mirroring classical multipole theory in physics by encoding atomic properties as monopole, dipole, and quadrupole tensors [25].

Comparative Model Architectures

Table 1: Architectural Characteristics of Leading Universal MLIPs

Model	Core Architectural Approach	Symmetry Handling	Force Prediction	Parameter Range
MACE	Graph neural network with higher-order body-ordered messages [27]	E(3)-equivariant [25]	Conservative (EFSG) [27]	~9.1M parameters [27]
Orbital (ORB)	Graph Network Simulator with smooth message updates [27]	Direct force prediction (non-conservative) or analytic gradient (conservative) [27]	Both conservative (ORB-v3c) and direct (ORB-v2, ORB-v3d) [27]	25-26M parameters [27]
eSEN (equivariant Smooth Energy Network)	Equivariant transformer with focus on smooth node representations [27]	E(3)-equivariant with smooth potential energy surfaces [62]	Primarily conservative (EFSG) [27]	~30M parameters [27] [71]
EquiformerV2 (eqV2)	Equivariant transformer with computational efficiency improvements [27]	E(3)-equivariant [27]	Direct force prediction (EFSD), non-conservative [27]	~87M parameters [27]

Performance Benchmarking and Comparative Analysis

Multi-Dimensional Accuracy Assessment

A comprehensive benchmark evaluating predictive capabilities across systems of varying dimensionality revealed that while all tested models demonstrate excellent performance for three-dimensional systems, accuracy degrades progressively for lower-dimensional structures [27]. The best performing models for geometry optimization were ORB-v2, EquiformerV2, and eSEN, with eSEN also providing the most accurate energies [27]. These models yield, on average, errors in atomic positions of 0.01–0.02 Å and errors in energy below 10 meV/atom across all dimensionalities [27].

Specialized Application Performance

Table 2: Performance Comparison Across Key Applications

Application Domain	Best Performing Model(s)	Key Performance Metrics	Reference Study
MOF Structure Optimization	PFP, eSEN-OAM, ORB-v3-omat+D3	92/100 structures within ±10% volume change vs DFT [71]	MOFSimBench [71]
MOF Molecular Dynamics	eSEN-OAM, PFP, ORB-v3-omat+D3	Highest number of structures with	ΔV	< 10% during NPT MD [71]	MOFSimBench [71]
MOF Bulk Modulus	eSEN-OAM, PFP	MAE for successful bulk modulus predictions [71]	MOFSimBench [71]
MOF Heat Capacity	PFP, ORB-v3-omat+D3, UMA	Lowest MAE for specific heat capacity at 300K [71]	MOFSimBench [71]
Surface Stability Prediction	OMat24-trained models (various architectures)	<6% MAPE on cleavage energies, 87% correct stable surface identification [72]	Zero-shot cleavage energy benchmark [72]
Biomolecular Systems	OrbitAll, MACE	MAE ~1.13 kcal/mol for HAT reactions in peptides [73] [74]	Peptide HAT reactions [74]

Training Data Dependence and Generalization

A critical finding across multiple studies is that strategic training data composition often delivers greater performance improvements than architectural sophistication. In a comprehensive zero-shot evaluation of 19 uMLIPs for predicting cleavage energies, models trained on the Open Materials 2024 (OMat24) dataset achieved mean absolute percentage errors below 6% despite never being explicitly trained on surface energies [72]. Architecturally identical models trained exclusively on equilibrium structures showed five-fold higher errors (15% MAPE), revealing that training data quality is 5–17 times more influential than model complexity for out-of-distribution generalization [72].

Experimental Protocols for MLIP Evaluation

Standardized Benchmarking Workflow

Diagram 1: MLIP Benchmarking Workflow

Key Experimental Methodologies

Structure Optimization Protocol

Structures are optimized using MLIPs and compared against DFT-optimized references. Performance is quantified by volume change rate (ΔV = 1 - V_MLIP/V_DFT), with successful predictions defined as |ΔV| < 10% [71]. This protocol typically employs 100+ diverse structures (MOFs, COFs, zeolites) to ensure statistical significance [71].

Molecular Dynamics Stability Assessment

After equilibration through optimization and NVT calculations, NPT simulations are run for 50 ps at 300K and 1 bar [71]. Stability is evaluated by volume change between initial and final structures (ΔV = 1 - V_final/V_initial), with |ΔV| < 10% indicating robust performance [71].

Cleavage Energy Calculation

Surface energies are computed by creating slabs of different Miller indices, calculating the energy difference between cleaved and bulk structures: E_cleavage = (E_slab - n × E_bulk)/2A, where n is the number of bulk units and A is the surface area [72]. This zero-shot evaluation tests generalization to properties outside training distributions.

Host-Guest Interaction Energy

For adsorption applications, interaction energies are computed as E_int = E_system - (E_host + E_guest), evaluating performance across repulsion, equilibrium, and weak-attraction regimes [71].

Critical Benchmark Datasets

Table 3: Essential Datasets for MLIP Training and Validation

Dataset	Domain Focus	Scale and Diversity	Primary Applications
Open Molecules 2025 (OMol25) [62]	Biomolecules, electrolytes, metal complexes	>100M calculations, ωB97M-V/def2-TZVPD level	High-accuracy molecular property prediction
Open Materials 2024 (OMat24) [27] [72]	Bulk materials with non-equilibrium configurations	Systematic perturbations, MD at extreme temperatures	Surface property prediction, out-of-distribution generalization
ODAC25 [75]	Metal-organic frameworks (MOFs)	~70M DFT calculations, 15,000 MOFs, 4 adsorbates	Sorbent discovery, host-guest interactions
MOFSimBench [71]	MOFs, COFs, zeolites	100 diverse structures, 5 tasks	Comprehensive MLIP evaluation across multiple properties
QMOF [71]	Metal-organic frameworks	~20,000 structures	Energy prediction accuracy assessment

Computational Tools and Frameworks

The autoplex framework provides automated implementation of iterative exploration and MLIP fitting through data-driven random structure searching, interfaced with widely-used computational infrastructure [2]. The atomate2 framework underpins high-throughput materials exploration, while torch-dftd enables dispersion correction in MLIP predictions [2] [71]. For large-scale molecular systems, tools like Schrödinger for protonation state sampling and Architector for metal complex generation are essential for dataset preparation [62].

Implications for Drug Development and Materials Discovery

The advancements in MLIP technology have profound implications for pharmaceutical research and materials science. For drug development professionals, models like OrbitAll demonstrate superior performance in predicting energies of charged, open-shell, and solvated molecules while robustly extrapolating to molecules significantly larger than training data [73]. This capability is crucial for accurate modeling of protein-ligand interactions, binding affinities, and reaction mechanisms in enzymatic environments [73].

In materials discovery, the ability of modern uMLIPs to efficiently simulate complex systems containing subsystems of mixed dimensionality opens new possibilities for modeling realistic materials and interfaces [27]. The consistent performance of leading models across structure optimization, molecular dynamics, and property prediction tasks suggests they have reached sufficient maturity to serve as direct replacements for DFT calculations in many applications, at a fraction of the computational cost [27] [71].

The field of machine learning interatomic potentials is rapidly evolving toward truly universal foundational models. Future developments will likely focus on strategic expansion of training data to cover poorly performing chemical systems (halogens, f-block elements) and low-symmetry structures [72]. Automated gap identification workflows that locate regions of chemical space with uncertain predictions will enable targeted training data generation [72]. Architectural innovations may increasingly prioritize inference speed alongside accuracy, with model distillation emerging as a key technique for knowledge transfer [62].

In conclusion, while architectural differences among MACE, ORB, eSEN, and EquiformerV2 contribute to their distinctive performance profiles, the training data composition has emerged as a dominant factor influencing model generalization. The research community now has access to multiple models with complementary strengths, enabling researchers to select architectures based on specific application requirements, computational constraints, and target accuracy thresholds. As these models continue to mature, they promise to accelerate the exploration of potential energy surfaces at unprecedented scales, fundamentally transforming computational approaches to materials design and drug development.

The hydrogen abstraction reaction, H + CH₄ → H₂ + CH₃, serves as a fundamental prototype for understanding polyatomic reaction dynamics. This reaction represents a critical benchmark system for theoretical chemistry, bridging the gap between simple atom-diatom reactions and the complex dynamics of real-world combustion processes. In the context of modern machine learning (ML) research, this reaction provides an ideal test case for developing and validating novel approaches to exploring potential energy surfaces (PESs). The accurate construction of a PES for this six-atom system presents significant computational challenges, making it an excellent target for the application of advanced ML techniques that can efficiently map the intricate relationship between molecular configuration and energy [4].

This case study examines how delta-machine learning (Δ-ML) methodologies have been successfully implemented to create high-level PESs for the H + CH₄ system, dramatically reducing computational costs while maintaining quantum-mechanical accuracy [4]. Furthermore, we explore how state-of-the-art experimental techniques provide crucial validation data, creating a feedback loop that continuously refines computational models. The integration of these computational and experimental approaches represents a paradigm shift in reaction dynamics, enabling unprecedented insights into kinetic and dynamic properties of complex chemical systems.

Machine Learning Approaches for Potential Energy Surface Development

Delta-Machine Learning Methodology

The Δ-ML framework has emerged as a powerful strategy for developing accurate PESs with significantly reduced computational expense. This approach leverages the strengths of both low-level and high-level quantum chemical calculations. As applied to the H + CH₄ system, the methodology follows a specific workflow [4]: First, a flexible analytical PES (PES-2008) serves as the low-level base model, enabling efficient sampling of configurations. A large number of points are sampled from this low-level surface. These configurations are then reevaluated using highly accurate quantum chemical methods, specifically the permutation invariant polynomial neural network (PIP-NN) surface. The key innovation lies in training the ML model to predict the difference (Δ) between the high-level and low-level energies, rather than the total energy itself. This Δ-ML PES effectively combines the broad coverage of the low-level model with the accuracy of the high-level method.

The validity of this approach was rigorously tested through comprehensive kinetic and dynamic studies [4]. Researchers performed variational transition state theory calculations with multidimensional tunneling corrections to analyze kinetics, and conducted quasiclassical trajectory calculations for the deuterated reaction H + CD₄ to explore dynamics. The results demonstrated that the Δ-ML approach faithfully reproduced the kinetics and dynamics information of the high-level PIP-NN surface, confirming its effectiveness in describing complex multidimensional polyatomic systems. This methodology represents a significant advancement, making high-accuracy dynamics studies computationally feasible for systems of this complexity.

Broader ML Applications in Combustion Kinetics

Beyond specific PES development, machine learning is revolutionizing combustion kinetics more broadly. Recent research has focused on developing universal ML methods to predict temperature-dependent rate constants across diverse reaction classes fundamental to combustion chemistry [76]. These approaches typically utilize reaction fingerprints derived from natural language processing of simplified molecular-input line-entry system (SMILES) strings, which effectively capture fine-grained differences between reaction classes [76]. Deep neural network models then use these fingerprints to predict the three modified Arrhenius parameters (ln A, n, and Ea), enabling the accurate reconstruction of complete temperature-dependent rate expressions [76].

This capability is particularly valuable for combustion modeling, where detailed kinetic mechanisms may involve tens of thousands of elementary reactions [76]. Traditional quantum chemical calculations become computationally prohibitive at this scale, creating a critical niche for ML approaches. By training on high-quality datasets derived from quantum chemistry for a subset of reactions, ML models can generalize to predict rate constants for similar reactions, dramatically accelerating model development while maintaining physical accuracy. This paradigm is transforming how researchers build comprehensive kinetic models for practical fuels and combustion systems.

Experimental Validation Techniques

State-Correlated Reaction Dynamics

Advanced experimental techniques provide the essential validation data for computational predictions. A groundbreaking methodology recently demonstrated for the analogous F + CH₄ → CH₃(v₁) + HF(v) reaction utilizes a three-dimensional velocity-map imaging detector with vacuum-ultraviolet photoionization [77]. This approach represents a significant advancement in universal detection with state-resolving capability. The power of this technique lies in its ability to simultaneously unveil both product vibrational branching and state-resolved angular distributions in a (v₁, v) pair-correlated manner from a single product-image measurement [77]. This provides previously inaccessible insights into the detailed quantum state correlations of reaction products.

The experimental data obtained through this method enabled direct comparison with six-dimensional quantum dynamics calculations, showing excellent agreements and thereby validating the theoretical approach [77]. Such state-correlated measurements are particularly valuable for identifying reactive resonances and other subtle quantum dynamical effects in polyatomic reactions. The general nature of this methodology opens new opportunities to gain deeper insights into many important complex chemical processes that have previously resisted detailed experimental characterization. For the H + CHâ‚„ reaction family, these experimental advances provide the critical benchmark data needed to validate the ML-derived PESs and resulting dynamics simulations.

Crossed Molecular Beam Experiments

Crossed molecular beam experiments with universal detection represent another powerful technique for probing reaction dynamics. These experiments typically employ electron bombardment ionization or photoionization mass spectrometry coupled with product time-of-flight measurements [77]. While these detection schemes have played pivotal roles in advancing our understanding of chemical reactions, they traditionally lack product state-specific information. The recent integration of velocity-map imaging detectors with vacuum-ultraviolet photoionization probes has overcome this limitation, creating a versatile experimental platform that combines universality with state-specific resolution [77].

The experimental setup typically involves crossing well-collimated, quantum-state-selected beams of reactants under high vacuum conditions to ensure single collision conditions. The resulting products are then ionized by carefully tuned vacuum-ultraviolet radiation and accelerated onto a position-sensitive detector. The resulting images contain complete information about the speed and angular distributions of the reaction products, which can be inverted to obtain the differential cross sections in the center-of-mass frame. When coupled with time-sliced ion imaging techniques, this approach provides unprecedented detail about the quantum-state-resolved dynamics of prototypical reactions like H + CHâ‚„.

Computational Protocols and Methodologies

First-Principles Molecular Dynamics

First-principles molecular dynamics (FPMD) based on density functional theory provides another computational approach for studying reaction mechanisms, particularly for complex combustion systems. In a recent study of CH₄/air mixtures combustion, FPMD simulations were employed to simulate the reaction of CH₄ and O₂ at constant temperatures of 3000 K and 4000 K [78]. The computational model contained 72 CH₄ molecules and 216 O₂ molecules (792 atoms total) in a cubic box, with dynamics based on the Born-Oppenheimer approximation [78]. Through cluster analysis and reaction tracking techniques, researchers identified 22 intermediates and 123 elementary reactions, including novel species such as HCOOH and O₃ not present in traditional combustion models [78].

This FPMD approach enabled the construction of a detailed chemical kinetic model (FP model), which was subsequently simplified using directed relation graph (DRG) and computational singular perturbation (CSP) methods to produce a reduced model (R-FP) containing only 20 species and 30 reactions [78]. This reduced model maintained predictive accuracy while being computationally efficient enough for complex multi-dimensional combustion simulations. The success of this "first-principles model construction + model simplification + engineering verification" scheme demonstrates the power of combining high-level theoretical methods with practical engineering considerations [78].

Automated Potential Exploration Frameworks

The increasing complexity of MLIP development has spurred efforts to automate the entire process of potential energy surface exploration and fitting. Recently, the autoplex framework ("automatic potential-landscape explorer") has been introduced as an openly available software package for this purpose [2]. This automated system implements iterative exploration and MLIP fitting through data-driven random structure searching, significantly reducing the human effort required to develop robust potentials [2].

The autoplex framework is particularly designed for interoperability with existing software architectures and enables high-throughput MLIP creation on high-performance computing systems [2]. In benchmark tests, the system successfully produced accurate potentials for diverse systems including the titanium-oxygen system, SiOâ‚‚, crystalline and liquid water, and phase-change memory materials [2]. While current benchmarks focus on bulk systems, the methodology illustrates how automation can accelerate atomistic machine learning in computational materials science, potentially including the development of PESs for reactive systems like H + CHâ‚„.

Data Presentation and Analysis

Kinetic Parameters for H + CHâ‚„ and Isotopologues

Table 1: Comparative Kinetic Parameters for Hydrogen Abstraction Reactions

Reaction	Methodology	Rate Constant Expression	Temperature Range (K)	Tunneling Correction
H + CH₄ → H₂ + CH₃	Δ-ML PES with VTST	To be determined from dynamics calculations	300-2500	Multidimensional
H + CD₄ → HD + CD₃	Quasiclassical Trajectories on Δ-ML PES	Product branching ratios and angular distributions	N/A	N/A
F + CH₄ → HF + CH₃	State-Correlated Imaging	Product pair correlation matrices	Crossed beam conditions	Quantum dynamical

Performance Metrics for Computational Methods

Table 2: Accuracy and Efficiency of Computational Approaches

Method	Computational Cost	Accuracy for H + CHâ‚„	Key Advantages	Limitations
Δ-ML from Analytical PES	Moderate (~10-100× cheaper than full quantum)	Reproduces high-level kinetics and dynamics [4]	Cost-effective for high-level dynamics	Dependent on quality of base PES
Direct Dynamics with MLIP	High for training, low for application	Quantitative for targeted systems [2]	No explicit PES parameterization	Requires extensive training data
First-Principles MD (DFT)	Very High	Reveals novel intermediates and pathways [78]	No preconceived mechanism	Limited to short timescales
Universal ML Rate Prediction	Low after training	Accurate across multiple reaction classes [76]	Broad applicability	Limited extrapolation beyond training

Research Reagent Solutions and Computational Tools

Table 3: Essential Research Tools for Reaction Dynamics Studies

Tool/Reagent	Function/Role	Specific Application Example
Potential Energy Surface (PES)	Defines energy as a function of nuclear coordinates	Δ-ML PES for H + CH₄ reaction [4]
Permutation Invariant Polynomial Neural Network (PIP-NN)	Provides high-level reference data for ML training	Accurate PES for H + CHâ‚„ [4]
Velocity-Map Imaging Detector	Measures product velocity and angular distributions	State-correlated dynamics in F + CHâ‚„ [77]
Vacuum-Ultraviolet Photoionization Probe	State-selective detection of reaction products	Universal detection with state resolution [77]
Directed Relation Graph (DRG)	Mechanism reduction for complex kinetic models	Simplifying detailed combustion mechanisms [78]
Computational Singular Perturbation (CSP)	Time-scale analysis for kinetic model reduction	Creating reduced models for engineering [78]
Reaction Fingerprints (SMILES-based)	Representing chemical reactions for ML	Predicting rate constants across reaction classes [76]
autoplex Software Package	Automated exploration and fitting of PES	High-throughput MLIP development [2]

Workflow and Signaling Pathways

Delta-Machine Learning Workflow for PES Development

Experimental Validation Pathway for Reaction Dynamics

The integration of machine learning approaches with high-level theoretical dynamics and state-of-the-art experiments has transformed our ability to study prototypical reactions like H + CH₄. The Δ-ML methodology has proven particularly effective for developing accurate PESs at manageable computational cost, enabling detailed kinetics and dynamics studies that were previously infeasible [4]. Concurrent advances in experimental techniques, especially state-correlated velocity imaging, provide the essential validation data needed to benchmark these computational approaches [77].

Looking forward, the increasing automation of PES exploration through frameworks like autoplex promises to further accelerate research in this field [2]. As ML methodologies continue to mature, we anticipate more generalized approaches that can handle increasingly complex reaction systems with minimal human intervention. The ongoing development of comprehensive, high-quality datasets will be crucial for training these next-generation models [76]. For the specific case of H + CHâ‚„, future work will likely focus on extending the accuracy of current methods to even more challenging regimes, including non-adiabatic effects and extended temperature and pressure ranges relevant to practical combustion environments.

The exploration of potential energy surfaces (PES) is fundamental to understanding molecular behavior, from chemical reactions to material properties. Machine learning (ML) has revolutionized this field by enabling large-scale, quantum-mechanically accurate atomistic simulations [2]. However, a significant challenge persists in the robust sampling and accurate representation of challenging regions of the PES, such as dissociation pathways and high-energy excited states. These areas are critical for modeling rare events and non-adiabatic processes but are often underrepresented in training datasets. This whitepaper assesses the performance of modern ML-driven frameworks in these demanding contexts, detailing specialized methodologies and reagents required for success.

Performance Challenges in Complex Configurations

The accuracy of Machine-Learned Interatomic Potentials (MLIPs) is not uniform across the entire potential-energy landscape. Performance can degrade significantly in regions far from equilibrium or with complex electronic structure.

Table 1: Performance of GAP-RSS Models for Different Systems

System / Phase	Target Accuracy (eV/atom)	DFT Single-Point Evaluations Required	Notable Challenges
Elemental Silicon (Si) [2]
Diamond-type	0.01	~500	High symmetry, well-described.
β-tin-type	0.01	~500	Slightly higher error than diamond-type.
oS24 allotrope	0.01	Few thousand	Lower-symmetry, metastable phase.
Binary Oxide (TiOâ‚‚) [2]
Rutile & Anatase	0.01	Achieved	Common polymorphs, accurately learned.
Bronze-type (TiOâ‚‚-B)	0.01	>1,000	Complex connectivity of polyhedra.
Full Binary System (Ti-O) [2]
Ti₂O₃, TiO, Ti₂O	0.01	>1,000 (varies by phase)	Diverse stoichiometries and electronic structures.

The data in Table 1 illustrates that achieving high accuracy for metastable phases (e.g., oS24 silicon) or phases with complex structural motifs (e.g., TiOâ‚‚-B) requires substantially more training data than for simpler, high-symmetry structures [2]. Furthermore, a model trained only on a single stoichiometry, such as TiOâ‚‚, fails catastrophically when applied to other compositions in the same system (e.g., rocksalt-type TiO), with errors exceeding 1 eV/atom [2]. This underscores the necessity for broad, system-wide sampling to create a truly robust potential.

Methodologies for Exploring Challenging Regions

Manually generating data for these rare events is a major bottleneck. Automated, iterative frameworks and active learning strategies are essential for efficient exploration.

Automated Framework for PES Exploration

The autoplex framework automates the exploration and fitting of PES, integrating high-throughput computing with active learning [2]. Its workflow, depicted below, is designed to minimize manual intervention while ensuring comprehensive sampling.

The process begins with Random Structure Searching (RSS) to generate diverse initial configurations [2]. These structures undergo DFT Single-Point Evaluation to create quantum-mechanical reference data. An MLIP (e.g., a Gaussian Approximation Potential, GAP) is then trained on this data [2]. A critical step is Active Learning, where the model's own uncertainty estimates are used to identify and query new, informative configurations (the "out-of-confidence" region) for DFT calculation, which are then added to the training set [79]. This iterative loop continues until the model achieves target accuracy across a range of test structures.

Protocol for Excited-State Dissociation Dynamics

The study of formaldehyde's H-atom dissociation on the lowest triplet state (T₁) provides a specific protocol for applying these methods to excited-state dynamics [79].

Experimental Workflow:

• PES Construction: A high-dimensional ML-PES for the T₁ state is constructed using an atomic-energy based deep neural network. The model uses weighted atom-centered symmetry functions (wACSFs) as inputs to satisfy physical symmetries [79].
• Active Learning & Validation:
  • Clustering: A clustering algorithm is applied to the training dataset to improve data efficiency [79].
  • Committee Models: Multiple NN models are trained; significant disagreement in their predictions flags geometries in the "out-of-confidence" region for additional ab initio calculation [79].
  • Benchmarking: The final NN-PES is validated against intrinsic reaction coordinate (IRC) pathways and small-scale trial dynamics from direct ab initio calculations [79].
• Dynamics Simulations: After validation, thousands of quasi-classical molecular dynamics trajectories are run on the ML-PES at a low computational cost. This allows for the investigation of mode-specific vibrational excitations on the dissociation probability [79].

The Scientist's Toolkit: Essential Research Reagents

Success in this field relies on a suite of specialized software tools and computational methods.

Table 2: Key Research Reagent Solutions

Item / Tool	Function & Explanation
autoplex [2]	An open-source software package implementing an automated framework for exploring and fitting PES. It integrates with high-throughput workflow systems (e.g., atomate2) to streamline MLIP development.
Gaussian Approximation Potential (GAP) [2]	A data-efficient MLIP framework based on Gaussian process regression, often used with the autoplex framework to drive RSS and potential fitting.
Active Learning (Uncertainty Quantification) [79]	A methodology where the ML model identifies regions of the PES where its prediction is uncertain. These "out-of-confidence" structures are targeted for new ab initio calculations, making data generation efficient.
Weighted Atom-Centered Symmetry Functions (wACSFs) [79]	A type of descriptor that converts atomic Cartesian coordinates into a fixed-length vector that is invariant to translation, rotation, and permutation of like atoms. Essential for representing the chemical environment to the NN.
Committee (or Quorum) of Models [79]	A technique where several ML models are trained independently. The variance in their predictions for a new structure serves as a measure of uncertainty, guiding active learning.
Quasi-Classical Molecular Dynamics	A dynamics method where the nuclei are treated as classical particles, but the initial conditions are quantized for vibrations. Used to simulate reaction dynamics, like H-atom dissociation, on the ML-PES [79].

The exploration of challenging regions on potential energy surfaces, such as dissociation limits and excited states, is now tractable through automated machine-learning frameworks. The key to success lies in implementing robust active learning protocols to ensure models are trained on data that adequately represents these complex and high-energy configurations. Tools like autoplex and methodologies built on uncertainty quantification are pushing the boundaries, enabling reliable and large-scale simulations of rare events that were previously prohibitive. This progress is critical for advancing research in catalysis, drug development, and materials science.

Conclusion

The integration of machine learning with potential energy surface exploration marks a paradigm shift in computational science, offering a powerful path to quantum-mechanical accuracy at a fraction of the computational cost. The key takeaways underscore the maturity of automated frameworks for robust PES development, the critical importance of high-quality and diverse data, and the emergence of universal models capable of handling systems from isolated molecules to complex interfaces. For biomedical and clinical research, these advances promise to dramatically accelerate drug discovery by enabling large-scale, accurate simulations of drug-target interactions, reaction mechanisms, and biomolecular dynamics that were previously infeasible. Future progress hinges on developing more data-efficient and interpretable models, improving generalizability across the entire chemical space, and seamlessly integrating these tools into multi-scale simulation workflows to tackle the complex challenges of modern therapeutics development.

References

• 1. Δ-machine learning from analytical functional forms [https://pubs.rsc.org/en/content/articlehtml/2025/cp/d5cp01980j]
• 2. An automated framework for exploring and learning ... [https://www.nature.com/articles/s41467-025-62510-6]
• 3. From the Automated Calculation of Potential Energy Surfaces ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC10961845/]
• 4. Δ-machine learning from analytical functional forms [https://pubs.rsc.org/en/content/articlelanding/2025/cp/d5cp01980j]
• 5. Visualising energy landscapes through manifold learning [https://arxiv.org/abs/2111.07843]
• 6. Machine Learning Speeds Up Quantum Chemistry Calculations [https://lindecenter.caltech.edu/news/machine-learning-speeds-quantum-chemistry-calculations]
• 7. Molecular set representation learning [https://www.nature.com/articles/s42256-024-00856-0]
• 8. QM40, Realistic Quantum Mechanical Dataset for Machine ... [https://www.nature.com/articles/s41597-024-04206-y]
• 9. An automated framework for exploring and learning ... [https://arxiv.org/html/2412.16736v1]
• 10. Janine George - autoatml/autoplex [https://www.linkedin.com/posts/janine-george-5b9027156_github-autoatmlautoplex-code-for-automated-activity-7282437900120555520-R6cR]
• 11. An implementation of artificial neural-network potentials for atomistic materials simulations: Performance for TiO2 - ScienceDirect [https://www.sciencedirect.com/science/article/abs/pii/S0927025615007806]
• 12. Deep-learning framework for fully-automated recognition of TiO2 polymorphs based on Raman spectroscopy | Scientific Reports [https://www.nature.com/articles/s41598-022-26343-3]
• 13. Evaluation of machine learning models for predicting TiO2 ... [https://www.nature.com/articles/s41598-024-64486-7]
• 14. The role of machine learning in predicting titanium dioxide ... [https://www.sciencedirect.com/science/article/abs/pii/S0304389425011550]
• 15. Substrate-aware computational design of two-dimensional ... [https://www.nature.com/articles/s41524-025-01754-8]
• 16. Accurate machine learning force fields via experimental ... [https://www.nature.com/articles/s41524-024-01251-4]
• 17. Design, Assessment, and Application of Machine Learning ... [https://arxiv.org/html/2511.00951v1]
• 18. An automated framework for exploring and learning ... [https://pubmed.ncbi.nlm.nih.gov/40825769/]
• 19. Deep Neural Networks and Kernel Regression Achieve ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC6984975/]
• 20. A neural kernel method for capturing multiscale high- ... [https://www.sciencedirect.com/science/article/abs/pii/S0045782523004413]
• 21. Machine Learning vs. Neural Networks: Differences & ... [https://www.carmatec.com/blog/machine-learning-vs-neural-networks-key-differences-and-future-trends/]
• 22. A comparison between machine and deep learning models ... [https://www.nature.com/articles/s41598-024-70341-6]
• 23. Leading AI-Driven Drug Discovery Platforms [https://www.sciencedirect.com/science/article/abs/pii/S0031699725075118]
• 24. Machine Learning in Drug Discovery Market 2025 Trends [https://www.towardshealthcare.com/insights/machine-learning-in-drug-discovery-market-sizing]
• 25. A critical review of machine learning interatomic potentials ... [https://www.oaepublish.com/articles/jmi.2025.17]
• 26. Universal machine learning interatomic potentials are ... [https://www.nature.com/articles/s41524-025-01650-1]
• 27. Universal Machine Learning Potential for Systems with ... [https://arxiv.org/html/2508.15614v1]
• 28. Efficient equivariant model for machine learning ... [https://www.nature.com/articles/s41524-025-01535-3]
• 29. Universal machine learning potentials break dimensional ... [https://www.eurekalert.org/news-releases/1105026]
• 30. Universal Machine Learning Potentials under Pressure [https://arxiv.org/html/2508.17792v1]
• 31. DeepChem Equivariant: SE(3) [https://arxiv.org/html/2510.16897v1]
• 32. Basic stability tests of machine learning potentials for ... [https://arxiv.org/abs/2503.11537]
• 33. A comprehensive benchmark of active learning strategies ... [https://www.nature.com/articles/s41598-025-24613-4]
• 34. Active Learning in Machine Learning Guide [Full Guide] [https://encord.com/blog/active-learning-machine-learning-guide/]
• 35. The present state and challenges of active learning in drug ... [https://pubmed.ncbi.nlm.nih.gov/38642700/]
• 36. Deep Batch Active Learning for Drug Discovery [https://elifesciences.org/reviewed-preprints/89679]
• 37. Molecular optimization using a conditional transformer for ... [https://www.nature.com/articles/s42004-025-01437-x]
• 38. Active Learning Methods for Efficient Data Utilization and ... [https://arxiv.org/abs/2504.16136]
• 39. - Δ and Force Fields Machine Learned Potential Energy Surfaces [https://pubmed.ncbi.nlm.nih.gov/36527383/]
• 40. Transformative Role of Artificial Intelligence in Drug ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC12406033/]
• 41. Machine learning meets quantum mechanics in catalysis [https://www.frontiersin.org/journals/quantum-science-and-technology/articles/10.3389/frqst.2023.1232903/full]
• 42. Enhancing Activation Energy Predictions under Data ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11815826/]
• 43. [2011.11601] $ Δ $- Machine for Learning ... Potential Energy Surfaces [https://arxiv.org/abs/2011.11601]
• 44. AI in Drug Discovery 2025: Accelerating Pharma Innovation [https://kanerika.com/blogs/ai-in-drug-discovery/]
• 45. Machine Learning to Predict Potential Energy Surface of ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11571027/]
• 46. Molecular dynamics simulations and drug discovery [https://bmcbiol.biomedcentral.com/articles/10.1186/1741-7007-9-71]
• 47. Large language model guided automated reaction pathway ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC12375046/]
• 48. Molecular Dynamics Simulations and Drug Discovery - PMC [https://pmc.ncbi.nlm.nih.gov/articles/PMC10705576/]
• 49. A Record-Breaking Dataset to Train AI Models has Launched [https://newscenter.lbl.gov/2025/05/14/computational-chemistry-unlocked-a-record-breaking-dataset-to-train-ai-models-has-launched/]
• 50. [2503.04070] A Foundational Potential Energy Surface Dataset for Materials [https://arxiv.org/abs/2503.04070]
• 51. Quality Machine Learning Training Data [https://www.cloudfactory.com/training-data-guide]
• 52. The QCML dataset, Quantum chemistry reference data ... [https://www.nature.com/articles/s41597-025-04720-7]
• 53. Using machine learning to go beyond potential energy surface benchmarking for chemical reactivity | Nature Computational Science [https://www.nature.com/articles/s43588-023-00549-5]
• 54. 2025 Guide to Cloud Cost Optimization for Modern ... [https://www.uscloud.com/blog/cloud-cost-optimization-2025-guide/]
• 55. Coverage bias in small molecule machine learning [https://www.nature.com/articles/s41467-024-55462-w]
• 56. Molecular property prediction in the ultra‐low data regime [https://www.nature.com/articles/s42004-025-01592-1]
• 57. Exploring Dimensionality Reduction for Chemical Space ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11733715/]
• 58. Geometric deep learning for molecular property predictions ... [https://jcheminf.biomedcentral.com/articles/10.1186/s13321-024-00895-0]
• 59. Discrepancies and error evaluation metrics for machine ... [https://www.nature.com/articles/s41524-023-01123-3]
• 60. An automated framework for exploring and learning ... [https://arxiv.org/abs/2412.16736]
• 61. Asparagus: A toolkit for autonomous, user-guided ... [https://www.sciencedirect.com/science/article/pii/S0010465524003692]
• 62. Exploring Meta's Open Molecules 2025 (OMol25) & Universal ... [https://www.rowansci.com/blog/exploring-open-molecules-2025]
• 63. EMFF-2025: a general neural network potential for ... [https://www.nature.com/articles/s41524-025-01809-w]
• 64. Ten quick tips for ensuring machine learning model validity [https://pmc.ncbi.nlm.nih.gov/articles/PMC11412509/]
• 65. Δ-machine learning from analytical functional forms [https://chemrxiv.org/engage/chemrxiv/article-details/67fd2ef081d2151a02773f72]
• 66. Universal Machine Learning Potential for Systems with ... [https://arxiv.org/abs/2508.15614]
• 67. Massive Atomic Diversity: a compact universal dataset for ... [https://www.nature.com/articles/s41597-025-06109-y]
• 68. BOOM: Benchmarking Out-Of-distribution Molecular ... [https://arxiv.org/html/2505.01912v1]
• 69. the Matbench test set and Automatminer reference algorithm [https://www.nature.com/articles/s41524-020-00406-3]
• 70. Chemical shifts in molecular solids by machine learning [https://www.nature.com/articles/s41467-018-06972-x]
• 71. High-Accuracy and High-Speed MOF Calculations with ... [https://matlantis.com/en/resources/blog/mofsimbench_matlantis/]
• 72. Can recent machine learning interatomic potentials reliably ... [https://www.eurekalert.org/news-releases/1105029]
• 73. OrbitAll: A Unified Quantum Mechanical Representation ... [https://arxiv.org/html/2507.03853v1]
• 74. Learning Potential Energy Surfaces of Hydrogen Atom ... [https://arxiv.org/abs/2508.00578]
• 75. The Open DAC 2025 Dataset for Sorbent Discovery in ... [https://arxiv.org/html/2508.03162v1]
• 76. A machine learning method to predict rate constants for ... [https://www.sciencedirect.com/science/article/abs/pii/S0010218024000841]
• 77. State-correlated reaction dynamics unveiled in full from a ... [https://www.nature.com/articles/s41467-025-66587-x]
• 78. Detailed and reduced chemical kinetic model of CH4/air ... [https://www.sciencedirect.com/science/article/abs/pii/S0009250923007157]
• 79. Theoretical studies on triplet-state driven dissociation of ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC8654486/]