Surrogate-Based Optimization in Process Systems Engineering: A Guide for Biomedical Researchers

Abstract

This article provides a comprehensive exploration of surrogate-based optimization (SBO) techniques, tailored for researchers, scientists, and professionals in drug development and biomedical engineering. It covers the foundational principles of SBO as a solution for computationally expensive black-box problems common in process systems engineering. The scope extends to a detailed review of state-of-the-art methodologies, including Bayesian Optimization, deep learning surrogates, and ensemble methods, with specific applications in pharmaceutical process systems and prosthetic device design. The content further addresses critical challenges such as data scarcity and model reliability, offers comparative performance assessments of various algorithms, and concludes with future directions for integrating these powerful optimization techniques into biomedical and clinical research to accelerate innovation.

What is Surrogate-Based Optimization? Core Principles and Relevance to Biomedical Engineering

Surrogate-Based Optimization (SBO) has emerged as a powerful methodology for solving optimization problems where the objective function and/or constraints are computationally expensive to evaluate, poorly understood, or treated as a black-box system [1]. In process systems engineering, such challenges frequently arise when dealing with complex physics-based simulations (e.g., computational fluid dynamics), laboratory experiments, or large-scale process models [1]. The core principle of SBO is to approximate these expensive costly black-box functions with computationally cheap surrogate models–often called metamodels–which are then used to guide the optimization search efficiently [1] [2].

This approach is particularly valuable in data-driven optimization contexts, where derivative information is unavailable or unreliable, a category often referred to as derivative-free optimization (DFO) [1]. By constructing accurate surrogates from a limited set of strategically sampled data points, SBO algorithms can find optimal solutions with far fewer evaluations of the true expensive function, making them indispensable for modern engineering research and drug development where simulations or experiments are time-consuming and resource-intensive [3].

Mathematical Foundation and Key Algorithms

Problem Formulation

A generic unconstrained optimization problem can be formulated as shown in Equation 1, where the goal is to minimize an objective function ( f(\mathbf{x}) ) that depends on design variables ( \mathbf{x} ) within a feasible region ( \mathcal{X} \subseteq \mathbb{R}^{n_{x}} ) [1].

[ \min{\mathbf{x}} f(\mathbf{x}) \quad \text{subject to} \quad \mathbf{x} \in \mathcal{X} \subseteq \mathbb{R}^{n{x}} ]

In real-world applications, this formulation often extends to include constraints, making the problem even more challenging. The objective function ( f ) is frequently treated as a black box, meaning its analytical form is unknown, and we can only observe its output for given inputs [1]. Evaluating this function is typically computationally expensive, creating the need for efficient optimization strategies like SBO.

The SBO Workflow

The standard SBO workflow involves these key iterative steps:

• Initial Sampling: Select an initial set of points ( \mathbf{x}1, \mathbf{x}2, ..., \mathbf{x}_n ) using a Design of Experiments (DoE) approach and evaluate the expensive true function ( f ) at these points [2].
• Surrogate Model Construction: Fit a surrogate model ( \hat{f} ) to the collected data ( {(\mathbf{x}i, f(\mathbf{x}i))} ).
• Infill Criterion: Use the surrogate to propose new promising candidate point(s) ( \mathbf{x}_{new} ) by optimizing an acquisition function.
• Model Update: Evaluate ( f(\mathbf{x}_{new}) ), add the new data to the training set, and update the surrogate model.
• Termination: Repeat steps 3-4 until a convergence criterion or evaluation budget is met [1] [2].

The following diagram illustrates this iterative workflow:

Multiple surrogate modeling and optimization strategies have been developed, each with strengths suited to different problem types. The table below summarizes key algorithms and their primary characteristics.

Table 1: Key Surrogate-Based Optimization Algorithms and Characteristics

Algorithm	Full Name	Surrogate Type	Key Features	Typical Use Cases
BO [1]	Bayesian Optimization	Gaussian Process (GP)	Provides uncertainty estimates; balances exploration vs. exploitation	Expensive black-box functions; hyperparameter tuning
TuRBO [1]	Trust Region Bayesian Optimization	Multiple local GPs	Uses trust regions for scalable high-dimensional optimization	High-dimensional problems
COBYQA [1]	Constrained Optimization by Quadratic Approximations	Quadratic approximation	Specifically designed for constrained problems	Optimization with explicit constraints
ENTMOOT [1]	Ensemble Tree Model Optimization Tool	Gradient-boosted trees	Handles mixed variable types well	Problems with categorical/continuous variables
SNOBFIT [1]	Stable Noisy Optimization by Branch and Fit	Local linear models	Robust to noisy function evaluations	Noisy experimental data

Applications in Process Systems and Pharmaceutical Engineering

SBO finds extensive applications across process systems engineering and pharmaceutical research, where first-principles models are complex and simulations require significant computational resources.

Process Systems Engineering Case Studies

In chemical engineering, SBO enables efficient optimization of process systems under various constraints. Case studies demonstrate its effectiveness for reactor control optimization and reactor design under uncertainty, where traditional derivative-based methods struggle due to the computational cost of high-fidelity simulations [1]. These applications often feature stochastic elements and high-dimensional parameter spaces, making SBO particularly valuable [1].

Another significant application is system architecture optimization (SAO) for complex engineered systems like jet engines [2]. These problems present challenges such as mixed-discrete design variables (e.g., choosing component types and continuous parameters), multiple objectives, and hidden constraints where simulations fail for certain design configurations [2]. SBO successfully navigates these complex design spaces while managing evaluation failures that can affect up to 50% of proposed points in some applications [2].

Pharmaceutical and Energy System Applications

While the search results focus on process and energy systems, the methodologies translate directly to pharmaceutical applications. For instance, SBO can optimize drug formulation parameters, bioreactor operation conditions, or pharmaceutical crystallization processes where experiments are costly and time-consuming.

In energy systems, a recent study applied data-driven surrogate optimization to deploy heterogeneous multi-energy storage at a building cluster level [4]. This approach addressed the challenge of optimally selecting and sizing different energy storage technologies (batteries, thermal storage) for individual buildings with highly diversified energy use patterns [4]. The method utilized genetic programming symbolic regression to develop accurate surrogate models and an iterative optimization with automated screening to handle the mixed combinatorial-continuous optimization problem [4]. This demonstrates how SBO can manage problems with both configuration selection and parameter sizing decisions.

Essential Protocols for SBO Implementation

Protocol 1: Handling Hidden Constraints in SAO

System architecture optimization often encounters hidden constraints–regions of the design space where function evaluations fail due to non-converging solvers or infeasible physics [2]. This protocol outlines a strategy for managing these constraints using Bayesian Optimization with a Probability of Viability (PoV) prediction.

Table 2: Reagents and Computational Tools for Hidden Constraint Management

Research Reagent	Function/Purpose	Implementation Notes
Mixed-Discrete Gaussian Process (MD-GP) [2]	Models objective function while handling both continuous and discrete variables	Essential for architectural decisions with both parameter types
Random Forest Classifier (RFC) [2]	Predicts failure regions and calculates Probability of Viability (PoV)	Can identify patterns leading to evaluation failures
Probability of Viability (PoV) Threshold [2]	Screening criterion for proposed infill points	Avoids evaluating points likely to violate hidden constraints
Ensemble Infill Strategy [2]	Generates multiple candidate points per iteration	Improves exploration while managing parallel evaluations

Procedure:

• Initialization: Generate an initial Design of Experiments (DoE) with ( n_{doe} ) points, ensuring they are spatially distributed across the design space [2].
• Viability Screening: Evaluate all initial points and identify which evaluations succeed (viable) and which fail.
• Model Building:
  • Train a mixed-discrete GP on viable points to model the objective function.
  • Train a classification model (e.g., Random Forest) to predict PoV based on all evaluated points.
• Infill Point Selection:
  • Generate candidate points by optimizing the acquisition function (e.g., Expected Improvement).
  • Apply the PoV threshold to filter out candidates with high failure probability.
  • Select the final infill points satisfying the PoV criterion.
• Iteration: Evaluate new points, update both models, and repeat until convergence.

The following diagram illustrates the hidden constraint handling strategy:

Protocol 2: Data-Driven Surrogate Optimization for Multi-Energy Storage Deployment

This protocol outlines a method for solving high-dimensional, nonlinear optimization problems using symbolic regression for surrogate modeling, particularly applicable to resource allocation problems with multiple technology options [4].

Procedure:

• Simulation Platform Development: Create a high-fidelity simulation model of the system (e.g., building cluster with multi-energy storage) to generate training data [4].
• Training Data Generation:
  • Define input variables (e.g., storage configuration choices, sizing parameters).
  • Run simulations across a wide range of input combinations.
  • Record corresponding performance metrics (e.g., energy costs, demand response performance).
• Surrogate Model Development:
  • Apply genetic programming symbolic regression to develop accurate, explicit surrogate models.
  • Validate model accuracy against holdout simulation data.
• Iterative Optimization with Screening:
  • Implement an optimization algorithm that uses the surrogate models.
  • Include automated option screening to efficiently navigate combinatorial choices.
  • Iteratively refine solutions while focusing computational resources on promising configurations.

Table 3: Performance Comparison of SBO Algorithms on Expensive Black-Box Functions

Algorithm	Problem Type	Key Performance Insight	Computational Efficiency
Bayesian Optimization (BO) [3]	General expensive black-box	Performance depends on evaluation time and available budget	Highly data-efficient for very expensive functions
Linear Surrogate SBO [5]	Airfoil self-noise minimization	Effective under small initial dataset constraints	Found design with 103.38 dB performance
CVAE Generative Approach [5]	Airfoil self-noise minimization	77.2% of generated designs outperformed SBO baseline	Provides diverse portfolio of high-performing candidates
Symbolic Regression SBO [4]	Multi-energy storage deployment	Reduced energy bills by 8%-181% vs. baseline cases	Handled high dimensionality effectively

The Researcher's Toolkit: Essential Software and Visualization

Key Software Tools and Libraries

Table 4: Essential Computational Tools for SBO Implementation

Tool/Library	Primary Function	Application Context
SBArchOpt [2]	Bayesian Optimization for system architecture problems	Handles mixed-discrete, hierarchical variables and hidden constraints
IDAES Surrogate Tools [6]	Surrogate model visualization and validation	Creates scatter, parity, and residual plots for model assessment
EXPObench [3]	Benchmarking library for expensive optimization problems	Standardized testing of SBO algorithms on real-world problems
ALAMO/PySMO [6]	Surrogate model training	Automated surrogate model generation from data
RS-51324	RS-51324, CAS:62780-15-8, MF:C11H11Cl2N3O2, MW:288.13 g/mol	Chemical Reagent
S-2474	S-2474|COX-2/5-LOX Inhibitor|CAS 158089-95-3

Surrogate Model Visualization and Validation

Effective SBO requires rigorous validation of surrogate model quality. The IDAES toolkit provides specialized plotting functions for this purpose [6]:

• Surrogate Scatter Plots: Visualize the relationship between individual inputs and outputs, comparing surrogate predictions against validation data.
• Parity Plots: Compare surrogate predictions against true function values across the validation set; ideal models show points closely aligned to the y=x line.
• Residual Plots: Analyze prediction errors (residuals) to identify systematic biases or regions where the surrogate performs poorly.

Implementation Code:

The Critical Need for SBO in Process Systems Engineering and Drug Development

Optimization is a cornerstone of modern engineering and science, impacting cost-effectiveness, resource utilization, and product quality across industries [7]. In complex chemical and pharmaceutical systems, traditional optimization methods that rely on analytical expressions and derivative information often fail when applied to problems involving computationally expensive simulators or experimental data collection [1]. This challenge has catalyzed the emergence of surrogate-based optimization (SBO) as a powerful methodology that combines machine learning with optimization algorithms to navigate expensive black-box problems efficiently [8].

SBO techniques approximate expensive functions through surrogate models trained on available data, dramatically reducing the number of costly evaluations required to find optimal solutions [1] [8]. For process systems engineering and drug development, where experiments or high-fidelity simulations can be prohibitively expensive and time-consuming, SBO provides a critical pathway to accelerate innovation while conserving resources [9]. This application note examines the transformative potential of SBO methodologies across these domains, providing structured protocols and frameworks for implementation.

Fundamental Principles of Surrogate-Based Optimization

Theoretical Foundation

SBO addresses optimization problems formulated as: $$\min{\mathbf{x}} f(\mathbf{x}), \quad \mathbf{x} \in \mathcal{X} \subseteq \mathbb{R}^{n{x}}$$ where $f$ represents an expensive-to-evaluate black-box function, and analytical expressions or derivative information are unavailable [1]. The core SBO approach replaces the expensive function $f(x)$ with a surrogate model $g(x)$ constructed from available data points using machine learning techniques [8].

The surrogate construction follows: $$\min{g} \sum{i=1}^{n} L(g(xi) - f(xi))$$ where $L$ represents a loss function between the surrogate predictions and actual function values [8]. This surrogate is then utilized within an acquisition function to determine promising new evaluation points: $$\arg \max_{x \in X} \alpha(g(x))$$ where $\alpha$ balances exploration against exploitation [8].

Algorithm Taxonomy

Table: Classification of Major SBO Algorithms

Algorithm Category	Representative Methods	Key Characteristics	Application Context
Bayesian Approaches	Bayesian Optimization (BO), TuRBO [7]	Uses probabilistic models; handles uncertainty effectively	High-dimensional problems with limited evaluations [7]
Local Approximation	COBYLA, COBYQA [7]	Constructs linear or quadratic local models	Low-dimensional constrained optimization [7]
Tree-Based Methods	ENTMOOT [7]	Uses decision trees as surrogates	Problems with structured input spaces [7]
Radial Basis Functions	DYCORS, SRBFStrategy [7]	Uses RBF networks as surrogates	Continuous black-box optimization [7]
Multimodal Frameworks	AMSEEAS [10]	Combines multiple surrogate models adaptively	Problems with complex response surfaces [10]

SBO in Process Systems Engineering

Chemical Process Optimization

In chemical engineering, SBO enables efficient optimization of processes where first-principles models are computationally demanding or where processes are guided purely by collected data [1]. Applications range from reactor control optimization to resource utilization improvement and sustainability metrics enhancement [7]. The digitalization of chemical engineering through smart measuring devices, process analytical technology, and the Industrial Internet of Things has further amplified the need for data-driven optimization approaches [1].

Sustainable Process Design

SBO contributes significantly to sustainable engineering through three interconnected dimensions:

• SBO for Sustainability: Application to sustainable engineering problems aligned with United Nations Sustainable Development Goals [8]
• Sustainability of SBO: Minimizing computational resources and energy consumption of optimization algorithms [8]
• Sustainability with SBO: Reducing expensive function evaluations of computationally intensive simulators [8]

Recent frameworks have demonstrated simultaneous improvements in multiple process metrics, including yield enhancement and process mass intensity reduction in pharmaceutical manufacturing [9].

SBO in Pharmaceutical Development

Drug Development Pipeline Optimization

The pharmaceutical sector increasingly depends on advanced process modeling to streamline drug development and manufacturing workflows [9]. SBO provides a practical solution for optimizing these complex systems while respecting stringent quality constraints.

A notable example is SPARC's development of SBO-154, an antibody-drug conjugate (ADC) for advanced solid tumors [11] [12] [13]. The successful completion of IND-enabling preclinical studies with favorable results demonstrates the potential of systematic optimization approaches in accelerating therapeutic development [13].

Pharmaceutical Process Optimization

Recent research has established novel SBO frameworks specifically designed for pharmaceutical process systems [9]. These frameworks integrate multiple software tools into unified systems for surrogate-based optimization of complex manufacturing processes, with demonstrated improvements in key metrics including:

• Yield improvement up to 3.63% in multi-objective optimization frameworks [9]
• Process Mass Intensity improvement of 7.27% in single-objective optimization [9]
• Maintenance of high purity levels while enhancing yield [9]

Integrated SBO Framework Protocol

Geometric Feature Knowledge-Driven Optimization

The aerodynamic supervised autoencoder (ASAE) framework provides a transferable methodology for leveraging domain knowledge in SBO [14]. This approach extracts features correlated with performance metrics to guide the optimization process more efficiently.

Diagram 1: Geometric feature knowledge-driven SBO workflow. This framework improves optimization efficiency by approximately twofold while achieving superior performance [14].

Adaptive Multi-Surrogate Protocol

The Adaptive Multi-Surrogate Enhanced Evolutionary Annealing Simplex (AMSEEAS) algorithm provides a robust methodology for time-expensive environmental and process optimization problems [10].

Experimental Protocol: AMSEEAS Implementation

• Initialization Phase

  • Define optimization problem: decision variables, objectives, constraints
  • Select ensemble of surrogate models (Kriging, Radial Basis Functions, Neural Networks)
  • Initialize population using Latin Hypercube Sampling
  • Evaluate initial samples using expensive simulator

• Iterative Optimization Phase

  • Step 1: Train all surrogate models on current dataset
  • Step 2: Calculate selection probabilities for each model based on recent performance
  • Step 3: Select active surrogate using roulette-wheel selection
  • Step 4: Generate candidate solutions using Evolutionary Annealing Simplex method
  • Step 5: Evaluate promising candidates using expensive simulator
  • Step 6: Update dataset and model performance metrics

• Termination Phase

  • Check convergence criteria (maximum evaluations, stability of solution)
  • Return optimal solution and performance history

This multimodel approach ensures flexibility against problems with varying geometries and complex response surfaces, consistently outperforming single-surrogate methods in benchmarking studies [10].

Research Reagent Solutions

Table: Essential Computational Tools for SBO Implementation

Tool Category	Specific Solutions	Function in SBO Workflow	Application Examples
Surrogate Models	Kriging, Radial Basis Functions, Neural Networks, Decision Trees [7]	Approximate expensive objective functions	ENTMOOT (tree-based) [7], SRBF (radial basis) [7]
Optimization Algorithms	Bayesian Optimization, COBYLA, TuRBO [7]	Navigate surrogate surfaces to find optima	High-dimensional reactor control [7], Pharmaceutical process optimization [9]
Expensive Simulators	CFD, HEC-RAS, Pharmaceutical process models [14] [10] [9]	Provide ground truth data for surrogate training	Aerodynamic design [14], Hydraulic systems [10], Drug manufacturing [9]
Feature Learning	Aerodynamic Supervised Autoencoder (ASAE) [14]	Extract performance-correlated features from design space	Airfoil and wing optimization [14]
Multi-Model Frameworks	AMSEEAS [10]	Adaptive surrogate selection for complex problems	Time-expensive environmental problems [10]

Performance Assessment Protocol

Benchmarking Methodology

Comprehensive SBO performance assessment requires standardized evaluation across multiple dimensions:

• Test Functions: Apply algorithms to diverse mathematical functions with known optima [7]
• Constraint Handling: Evaluate performance on both constrained and unconstrained problems [7]
• Dimensionality Scaling: Test scalability from low-dimensional to high-dimensional problems [7]
• Real-World Case Studies: Validate using chemical engineering and pharmaceutical applications [7] [9]

Quantitative Metrics

Table: SBO Performance Assessment Metrics

Performance Dimension	Quantitative Metrics	Interpretation
Convergence Efficiency	Number of expensive function evaluations to reach target objective	Lower values indicate superior performance
Solution Quality	Percentage improvement in objective function vs. baseline	Higher values indicate superior optimization
Computational Sustainability	Energy consumption and computational resources required	Lower environmental impact of optimization process
Robustness	Performance consistency across diverse problem types	Higher reliability across applications

Surrogate-based optimization represents a paradigm shift in addressing complex, expensive optimization challenges across process systems engineering and pharmaceutical development. By leveraging machine learning to construct efficient approximations of costly simulations and experiments, SBO enables accelerated innovation while conserving computational and experimental resources. The structured frameworks and protocols presented in this application note provide researchers with practical methodologies for implementing SBO across diverse domains, from sustainable process design to accelerated drug development. As SBO methodologies continue to evolve, their integration into industrial practice promises to enhance both the efficiency and sustainability of technological advancement across critical engineering and healthcare sectors.

Surrogate-based optimization has emerged as a pivotal technique in process systems engineering, particularly for tackling costly black-box problems where derivative information is unavailable or the evaluation of the underlying function is computationally expensive [7] [1]. This approach involves constructing approximate models, or surrogates, of complex systems based on data collected from a limited number of simulations or physical experiments. These surrogates are then used to drive optimization, significantly reducing computational burden [15]. The adoption of these techniques is accelerating the digital transformation in fields like pharmaceutical manufacturing, where they streamline drug development and manufacturing workflows, leading to substantial improvements in operational efficiency, cost reduction, and adherence to stringent product quality standards [15] [9]. This application note details the core advantages of surrogate-based optimization—computational efficiency, sensitivity analysis, and enhanced system insight—and provides detailed protocols for their implementation, framed within the context of process systems engineering research.

Computational Efficiency in Surrogate-Based Optimization

Computational efficiency is the most immediate advantage of surrogate-based optimization. In chemical and pharmaceutical engineering, high-fidelity models—such as those involving computational fluid dynamics, quantum mechanical calculations, or integrated process flowsheets—can be prohibitively time-consuming to evaluate, making direct optimization infeasible [1]. Surrogate models address this by acting as fast-to-evaluate proxies for these expensive simulations.

Mechanism of Efficiency Gains

The efficiency is achieved through a two-phase process. First, a surrogate model is trained on a carefully selected dataset of input-output pairs from the expensive high-fidelity model or physical process. Subsequently, the optimization algorithm operates on the surrogate model, which can be evaluated orders of magnitude faster than the original system [15] [16]. This decoupling allows for extensive exploration and exploitation of the design space without the constant computational cost of running the full simulation.

Quantitative Evidence from Case Studies

Empirical studies across process engineering confirm these efficiency gains. In a pharmaceutical manufacturing case study, a surrogate-based optimization framework was applied to an Active Pharmaceutical Ingredient (API) manufacturing flowsheet. The results, summarized in Table 1, demonstrate that the framework successfully identified process conditions that led to measurable improvements in key performance indicators, all while avoiding the computational cost of repeatedly running the full process model [15] [9].

Table 1: Performance Improvements in a Pharmaceutical Manufacturing Case Study Using Surrogate-Based Optimization

Optimization Type	Key Performance Indicator	Improvement	Reference
Single-Objective	Yield	+1.72%	[15]
Single-Objective	Process Mass Intensity	+7.27%	[15]
Multi-Objective	Yield	+3.63% (while maintaining high purity)	[15] [9]

Another case study involving the optimization of a wet granulation process using an autoencoder-based inverse design reported computational times averaging under 4 seconds for the optimization run, highlighting the dramatic speed-up achievable with these methods [16].

Protocol: Implementing Surrogate-Based Optimization for Process Design

This protocol outlines the steps for applying surrogate-based optimization to a chemical or pharmaceutical process, such as the API manufacturing flowsheet referenced in the case study [15].

Experimental Design and Data Collection

• Step 1: Define Optimization Objectives and Constraints. Clearly articulate the goal (e.g., maximize yield, minimize energy consumption) and identify all process constraints (e.g., purity must exceed a certain threshold, operating pressure must remain within a safe window).
• Step 2: Select Input Variables. Identify the key decision variables (e.g., temperature, catalyst concentration, flow rates) and their feasible ranges.
• Step 3: Generate Initial Training Data. Use a space-filling design of experiments (DoE), such as Latin Hypercube Sampling, to select an initial set of points within the input variable space. The number of initial points should be a multiple (typically 5-10x) of the number of input variables.
• Step 4: Run High-Fidelity Simulations/Experiments. Execute the expensive, high-fidelity process model or physical experiment at each of the points specified by the DoE. Record the corresponding output metrics (e.g., yield, purity, cost).

Surrogate Model Construction and Validation

• Step 5: Choose a Surrogate Model Type. Select an appropriate machine learning model based on the problem characteristics. Common choices include:
  • Gaussian Process Regression (Kriging): Excellent for uncertainty quantification, used in Bayesian Optimization [7] [1].
  • Gradient-Boosted Trees (e.g., via ENTMOOT): Handles complex, non-linear relationships and naturally manages input constraints [7] [17].
  • Random Forests: Robust and provides feature importance metrics, as demonstrated in the mandibular movement study [18].
  • Neural Networks: Suitable for very high-dimensional and non-linear problems [17] [16].
• Step 6: Train the Model. Use the input-output data collected in Steps 3-4 to train the selected surrogate model.
• Step 7: Validate Model Accuracy. Assess the predictive performance of the surrogate on a separate, held-out test dataset using metrics like R² or Mean Squared Error. If accuracy is insufficient, return to Step 3 to augment the training data or to Step 5 to select a different model.

Optimization and Iteration

• Step 8: Perform Optimization over the Surrogate. Use a suitable optimization algorithm (e.g., evolutionary algorithms, branch-and-bound for tree surrogates, or dedicated solvers like Gurobi with OMLT/Pyomo) to find the input values that optimize the objective function, subject to the defined constraints [17].
• Step 9: Validate and Update. Run the high-fidelity model at the optimal point identified by the surrogate. To improve the model, add this new data point to the training set and retrain the surrogate (iterative updating). Repeat until convergence or a computational budget is reached.

The following workflow diagram illustrates this iterative process.

Enabling Sensitivity Analysis and System Insight

Beyond finding an optimum, surrogate-based optimization provides a powerful pathway for sensitivity analysis and deeper system insight. The surrogate model itself becomes a source of knowledge about the process.

Quantitative Sensitivity Analysis

Once trained, the surrogate model can be interrogated to determine how sensitive the output is to changes in each input variable. For tree-based models like those used in ENTMOOT or Random Forests, techniques like Gini importance or permutation importance can rank variables by their influence on the objective function [18] [17]. In the study using mandibular movement signals, feature importance analysis from a Random Forest model revealed that "event duration, lower percentiles, central tendency, and the trend of MM amplitude were the most important determinants" for classifying hypopnea events [18]. This quantitative insight guides engineers toward the most critical process parameters.

Visualizing Trade-offs and the Design Space

In multi-objective optimization, surrogates are used to construct Pareto fronts, which visually represent the trade-offs between competing objectives [15]. For instance, a Pareto front can show how much product purity must be sacrificed to achieve a higher yield. This allows researchers and decision-makers to visually navigate the design space and select an optimal operating point that balances multiple criteria, such as yield, purity, and sustainability [15]. Advanced methods like autoencoder-based inverse design further enhance insight by performing dimensionality reduction, which allows for the visualization of complex, high-dimensional design spaces in two or three dimensions, thereby improving process understanding [16].

Protocol: Conducting Global Sensitivity Analysis with a Trained Surrogate

This protocol describes how to use a trained surrogate model to perform a global sensitivity analysis, identifying the most influential input variables in your process.

Model-Based Sensitivity Indices

• Step 1: Trained Surrogate Model. Begin with a fully trained and validated surrogate model from Section 3.2.
• Step 2: Generate Large Sample Set. Create a large number of random input vectors (e.g., 10,000+) uniformly distributed across the entire feasible input space.
• Step 3: Run Predictions. Use the surrogate model to predict the output for each of these input vectors.
• Step 4: Calculate Sensitivity Indices.
  • For Tree-Based Models (Random Forest, ENTMOOT): Use built-in feature importance metrics. These typically measure the total decrease in node impurity (e.g., Gini or Mean Squared Error) weighted by the number of samples reaching that node, averaged over all trees.
  • For Other Models (Gaussian Processes, Neural Networks): Calculate Sobol indices. This involves computing the variance of the conditional expectation of the output. The first-order Sobol index (Si) measures the main effect of each input variable, while the total-order index (STi) includes interaction effects.

Interpretation and Visualization

• Step 5: Rank Variables. Rank all input variables from highest to lowest based on their calculated importance index (e.g., Gini importance or total-order Sobol index).
• Step 6: Create a Bar Chart. Plot the normalized importance scores in a bar chart for clear visual comparison.
• Step 7: Analyze and Report. The variables with the highest scores have the greatest influence on the process output. This knowledge allows researchers to focus experimental and control efforts on these key parameters.

The diagram below maps the logical flow of this analytical process.

The Scientist's Toolkit: Key Research Reagents and Software

Successful implementation of surrogate-based optimization requires a suite of computational tools and models. The table below lists essential "research reagents" for scientists in this field.

Table 2: Essential Tools and Software for Surrogate-Based Optimization Research

Tool Name	Type	Primary Function	Reference
ENTMOOT	Software Tool	Multi-objective black-box optimization using gradient-boosted trees (Gurobi as solver).	[7] [17]
OMLT	Python Package	Represents machine learning models (NNs, trees) within the Pyomo optimization environment.	[17]
Bayesian Optimization (BO)	Algorithm/ Framework	Efficient global optimization for expensive black-box functions, using Gaussian processes.	[7] [1]
TuRBO	Algorithm	State-of-the-art variant of BO that scales to high-dimensional problems.	[7]
Random Forest	Algorithm/ Model	Robust surrogate model that also provides feature importance for sensitivity analysis.	[18]
Autoencoder	AI Model	Used for inverse design and dimensionality reduction to visualize complex design spaces.	[16]
High-Fidelity Process Model	Digital Reagent	The complex, computationally expensive simulation of the physical process being optimized.	[15] [16]
Saroaspidin B	Saroaspidin B, CAS:112663-68-0, MF:C25H32O8, MW:460.5 g/mol	Chemical Reagent	Bench Chemicals
Saroaspidin C	Saroaspidin C, CAS:112663-70-4, MF:C26H34O8, MW:474.5 g/mol	Chemical Reagent	Bench Chemicals

Surrogate-based optimization represents a paradigm shift in how complex engineering systems are designed and improved. By leveraging computationally efficient surrogate models, researchers can solve previously intractable optimization problems, as evidenced by successful applications in pharmaceutical manufacturing that led to tangible improvements in yield and process intensity [15] [9]. Furthermore, the analytical power of these models extends beyond finding a single optimum; they enable rigorous sensitivity analysis to identify key process drivers and provide visual tools like Pareto fronts to understand critical trade-offs between competing objectives [15] [18]. As the field progresses, the integration of advanced AI, such as autoencoders for inverse design [16], and robust software frameworks, like OMLT and ENTMOOT [17], will continue to deepen system insight and accelerate innovation across process systems engineering.

Optimization is fundamental to chemical engineering and pharmaceutical development, directly impacting cost-effectiveness, resource utilization, and product quality [7] [19]. The methods for achieving optimal decisions have undergone a significant evolution, shifting from traditional model-based approaches to modern data-driven paradigms. This transition has been particularly impactful in process systems engineering and drug development, where the ability to optimize complex, expensive-to-evaluate systems without explicit analytical models provides a distinct advantage [19] [1].

Traditional optimization relied on algebraic or knowledge-based expressions that could be optimized using derivative information [1]. However, the rise of digitalization, smart sensors, and the Industrial Internet of Things (IIoT) has generated abundant process data, creating a need for algorithms that can leverage this information directly [19] [1]. This gave rise to data-driven optimization, also known as derivative-free, zeroth-order, or black-box optimization [1]. In this context, "black-box" refers to systems where the objective and constraint functions are only available as outputs from experiments or complex simulations, making derivative information unavailable or unreliable [19]. This review traces this methodological evolution, framed within the context of surrogate-based optimization for process systems engineering, and provides application notes and protocols for drug development researchers.

The Paradigm Shift: From Traditional to Data-Driven Methods

Traditional Model-Based Optimization

The traditional decision-making approach in chemical engineering leverages first-principles models—algebraic or differential equations derived from physical laws—that are optimized using derivative information from their analytical expressions [1]. This approach requires a deep mechanistic understanding of the system to develop accurate, differentiable models.

Key Limitations:

• Model Development Cost: Creating accurate first-principles models is time-consuming and requires expert knowledge.
• Computational Expense: Solving complex models involving rigorous differential-algebraic equations can be computationally prohibitive.
• Inflexibility: These models struggle to adapt to systems with unknown mechanisms or high stochasticity.

The Rise of Data-Driven and Surrogate-Based Optimization

Data-driven optimization bypasses the need for explicit first-principles models. It treats the system as a "black box," using input-output data to guide the search for an optimum [19] [20]. A dominant subset of these methods is surrogate-based optimization (also known as model-based derivative-free optimization). These methods iteratively construct and optimize approximate models of the expensive black-box function [7] [1].

The catalyst for this shift includes:

• Data Abundance: Proliferation of data from sensors and process analytical technology [19] [1].
• Problem Complexity: Increased need to optimize systems where models are unknown, proprietary, or too expensive to run frequently [19].
• Algorithmic Advances: Development of robust algorithms for building and optimizing surrogates.

Table 1: Comparison of Traditional and Data-Driven Optimization Paradigms

Feature	Traditional Optimization	Data-Driven (Surrogate-Based) Optimization
Core Input	Analytical model expressions	Input-output data from experiments or simulations
Derivative Use	Uses analytical gradients	Derivative-free; uses only function evaluations
Model Basis	First-principles (white-box)	Surrogate models (black-box or grey-box)
Computational Focus	Solving the model	Minimizing number of expensive function evaluations
Ideal Application	Well-understood, differentiable systems	Expensive black-box problems with unknown mechanisms

Key Algorithmic Families in Data-Driven Optimization

Derivative-free optimization (DFO) algorithms can be categorized into three main families [19] [1].

Direct Search Methods

These methods directly compare function values without constructing a surrogate model. Examples include the Nelder-Mead (simplex) algorithm and pattern search. They are often simple to implement but may require more function evaluations for convergence [19] [1].

Finite-Difference Methods

These methods approximate gradients using function evaluations, enabling the use of gradient-based optimization algorithms. Examples include finite-difference BFGS and Adam. They bridge direct and model-based methods [1].

Model-Based (Surrogate-Based) Methods

This is the most prominent family for expensive black-box problems. It involves:

• Design of Experiments: Selecting points in the design space to run the expensive evaluation.
• Surrogate Model Construction: Fitting a computationally cheap model to the collected data.
• Surrogate Optimization: Optimizing the surrogate to propose new candidate points.
• Model Update: Re-evaluating the expensive function at promising points and updating the surrogate [7] [19].

Table 2: Key Surrogate-Based Optimization Algorithms and Their Applications

Algorithm	Description	Strengths	Common Use Cases in Pharma/PSE
Bayesian Optimization (BO)	Uses Gaussian processes to model the objective and an acquisition function to guide sampling [7] [1].	Handles noise naturally; provides uncertainty estimates.	Hyperparameter tuning, stochastic simulation optimization [7] [21].
Trust-Region Methods (e.g., Py-BOBYQA, CUATRO)	Constructs local polynomial models within a trust region that is adaptively updated [19].	Strong theoretical convergence guarantees.	Flowsheet optimization, real-time optimization [19].
Radial Basis Function (RBF) Methods	Uses RBFs as global surrogates (e.g., DYCORS, SRBF) [7].	Effective for global exploration.	Process design, reactor optimization [7] [19].
Tree-Based Methods (e.g., ENTMOOT)	Uses ensemble trees (like random forests) as surrogates [7].	Handles categorical variables and non-smooth functions.	Material and process design with mixed variable types.

The following diagram illustrates the logical relationships and historical evolution of these key optimization methodologies.

Application in Process Systems Engineering and Drug Development

Data-driven optimization addresses critical challenges in process systems engineering and Model-Informed Drug Development (MIDD), enabling more efficient and predictive design [22].

Pharmaceutical Process Design and Optimization

In drug development, surrogate-based optimization is used to streamline manufacturing process design. A key application is optimizing tablet manufacturing processes to control Critical Quality Attributes (CQAs), such as dissolution behavior [23].

Protocol 1: Surrogate Modeling for Tablet Dissolution Behavior Prediction

Objective: To develop a surrogate model for predicting dissolution behavior in tablet manufacturing, identifying critical process parameters for efficient process design [23].

Workflow Summary: The diagram below outlines the surrogate modeling workflow for linking manufacturing inputs to dissolution profiles.

Materials and Reagents:

• Active Pharmaceutical Ingredient (API): e.g., Paracetamol (200 mg per tablet) [23].
• Excipients: Lactose monohydrate (70 mg), Microcrystalline cellulose (30 mg) [23].
• Simulation Software: gPROMS for mechanistic flowsheet modeling [23].
• Programming Environment: Python with Scikit-learn for Random Forest regression [23].

Step-by-Step Methodology:

• Define Input Space: Identify variable input parameters P (e.g., material properties, granulation liquid-to-solid ratio, compression force) and their feasible ranges [23].
• Generate Data via Mechanistic Modeling: Execute the mechanistic flowsheet model in gPROMS with different combinations of P to generate corresponding dissolution profiles D_mech(t). This is the expensive "black-box" function evaluation [23].
• Parameterize Output: Fit the raw dissolution profile D_mech(t) to a Weibull model (or other suitable function) to extract key parameters (e.g., shape and scale factors). This converts a time-series profile into a manageable set of numbers [23].
• Train Surrogate Model: Using the input parameters P as features and the fitted Weibull parameters as targets, train a Random Forest regression model. This model learns the mapping P -> Weibull Parameters [23].
• Validate Model: Compare the dissolution profile D_surr(t) predicted by the surrogate model against profiles generated by the mechanistic model for a validation dataset to ensure accuracy [23].
• Optimization and Sensitivity Analysis: Use the trained, fast-to-evaluate surrogate model to run extensive sensitivity analyses or optimization routines to identify critical process parameters and optimal operating conditions [23].

Active Pharmaceutical Ingredient (API) Manufacturing

A unified surrogate-based optimization framework can drive substantial improvements in API manufacturing metrics like yield, purity, and sustainability [15].

Protocol 2: Multi-Objective Optimization of an API Manufacturing Process

Objective: To optimize an API manufacturing flowsheet for competing objectives (e.g., maximize yield and purity, minimize environmental impact) using surrogate-based methods [15].

Materials and Reagents:

• Dynamic System Model: A high-fidelity dynamic model of the API manufacturing process (e.g., in gPROMS or Aspen Plus) [15].
• Optimization Framework: Software tools for surrogate modeling (e.g., Python, MATLAB) and multi-objective optimization (e.g., NSGA-II).

Step-by-Step Methodology:

• Define Objectives and Decision Variables: Identify key performance indicators (e.g., Yield, Process Mass Intensity) and the process variables to be optimized [15].
• Design of Experiments (DoE): Use a space-filling DoE (e.g., Latin Hypercube Sampling) to define a set of decision variable combinations to simulate.
• Run High-Fidelity Simulations: Execute the dynamic process model for each combination from the DoE to generate the corresponding objective function values.
• Construct Surrogate Models: Train individual surrogate models for each objective using the simulation data.
• Multi-Objective Optimization: Apply a multi-objective evolutionary algorithm to the surrogates to generate a Pareto front, which reveals the trade-offs between competing objectives [15].
• Decision Making: Select the optimal operating point from the Pareto front based on project priorities. Validate the selected point with a final high-fidelity simulation.

The Scientist's Toolkit: Essential Research Reagents and Computational Tools

The following table details key computational and methodological "reagents" essential for conducting data-driven optimization studies in process systems engineering and drug development.

Table 3: Key Research Reagent Solutions for Data-Driven Optimization

Tool/Reagent	Type	Function in Optimization
Mechanistic Simulators(gPROMS, Aspen Plus)	Software	Serves as the "expensive black-box" to generate high-fidelity input-output data for training surrogates [23].
Gaussian Process Regression	Statistical Model	A core surrogate model in Bayesian Optimization; provides predictions with uncertainty estimates [7] [21].
Random Forest / Decision Trees	Machine Learning Model	A surrogate model that handles non-smooth functions and mixed variable types effectively [7] [23].
Trust-Region Algorithm	Optimization Framework	Manages the trade-off between global exploration and local exploitation by dynamically adjusting the region where the surrogate is trusted [19].
Radial Basis Functions (RBF)	Mathematical Function	Used to construct flexible global surrogate models that interpolate scattered data points [7] [19].
Latin Hypercube Sampling	Algorithm	An experimental design method to generate efficient, space-filling samples from the input parameter space for initial surrogate training [15].
Expected Improvement (EI)	Acquisition Function	In Bayesian Optimization, guides the selection of the next sample point by balancing prediction mean and uncertainty [7].
SB-204900	(2R,3S)-N-Methyl-3-phenyl-N-[(Z)-2-phenylvinyl]-2-oxiranecarboxamide	Research-use (2R,3S)-N-Methyl-3-phenyl-N-[(Z)-2-phenylvinyl]-2-oxiranecarboxamide. Study its role in synthesizing bioactive molecules. For Research Use Only. Not for human consumption.
SB 216763	SB 216763, CAS:280744-09-4, MF:C19H12Cl2N2O2, MW:371.2 g/mol	Chemical Reagent

The evolution from traditional to data-driven optimization represents a fundamental shift in how complex systems are designed and controlled. Surrogate-based optimization techniques have emerged as a powerful methodology for tackling expensive black-box problems prevalent in process systems engineering and pharmaceutical development. By leveraging input-output data to construct computationally efficient surrogate models, these methods enable the optimization of systems where first-principles models are difficult, expensive, or impossible to develop and use directly. As demonstrated in applications ranging from tablet manufacturing to API process intensification, this data-driven paradigm shortens development timelines, reduces costs, and enhances the robustness of process design, ultimately accelerating the delivery of innovative therapies to patients.

In process systems engineering, optimization serves as a cornerstone for enhancing cost-effectiveness, resource utilization, product quality, and sustainability metrics [7] [1]. The rise of digitalization, smart measuring devices, and sensor technologies has intensified the need for sophisticated data-driven optimization approaches [1]. This document establishes foundational protocols for properly structuring optimization problems, with particular emphasis on formulation within surrogate-based optimization frameworks where derivative information may be unavailable or computationally expensive to obtain [7] [1]. A well-formulated optimization problem precisely defines the decision levers (variables), the performance metrics (objectives), and the operational limits (constraints) that govern the system under investigation.

Table 1: Core Components of an Optimization Problem

Component	Definition	Role in Optimization	Examples in Process Engineering
Decision Variables	Quantities controlled by the optimizer to find the optimal solution [24] [25].	Define the search space of possible solutions.	Reactor temperature, catalyst concentration, flow rates [1].
Objective Function	A function to be minimized or maximized [26] [24] [25].	Defines the performance criterion for evaluating solutions.	Minimize production cost, maximize product yield, minimize energy consumption [7].
Constraints	Conditions that must be satisfied for a solution to be feasible [26] [24].	Delineate the boundaries of acceptable operating conditions.	Maximum pressure limits, minimum purity requirements, safety thresholds [26].

Fundamental Concepts and Mathematical Formulation

Decision Variables

Decision variables (DVs) represent the independent parameters that the optimizer adjusts to find the optimum. In chemical engineering, these often pertain to geometric, operational, or physical aspects of a system [24]. DVs can be continuous (able to take any value within a range, such as temperature) or discrete (limited to specific values or types, such as the number of reactors) [24]. A critical practice is to begin with the smallest number of DVs that still captures the essence of the problem, thereby reducing complexity and improving the likelihood of convergence to a meaningful solution [24]. Furthermore, linearly dependent variables that control the same physical characteristic should be avoided to prevent an ill-posed problem with infinite equivalent solutions [24].

Objective Function

The objective function, typically denoted by ( Z ) or ( f ), is a real-valued function that quantifies the goal of the optimization, whether it is to minimize cost or maximize efficiency [26] [25]. In a general mathematical sense, an optimization problem is formulated as shown in Equation 1 [24]:

[ \begin{align} \text{Minimize} \quad & f_{\text{obj}}(\mathbf{x}) \ \text{With respect to} \quad & \mathbf{x} \ \text{Subject to} \quad & g_{\text{lb}} \leq g(\mathbf{x}) \leq g_{\text{ub}} \ \quad & h(\mathbf{x}) = h_{\text{eq}} \end{align} ]

Here, ( \mathbf{x} ) is the vector of decision variables, ( f_{\text{obj}} ) is the objective function, ( g(\mathbf{x}) ) represents inequality constraints with lower and upper bounds, and ( h(\mathbf{x}) ) represents equality constraints [26] [24]. It is standard to frame problems as minimizations; a maximization problem can be converted by applying a negative sign to the objective function (e.g., maximize profit is equivalent to minimize negative profit) [24].

Constraints

Constraints define the feasible region by setting conditions that the decision variables must obey. They can be classified as:

• Inequality Constraints: Specify that a function of the variables must be greater than or less than a specified value (e.g., ( hj(\mathbf{x}) \geq dj )) [26].
• Equality Constraints: Require that a function of the variables must exactly equal a specified value (e.g., ( gi(\mathbf{x}) = ci )) [26].

A design satisfying all constraints is feasible, while one violating any constraint is infeasible [24]. Unlike variable bounds, which are strictly respected, optimizers may temporarily violate constraints during the search process to navigate the design space [24].

Surrogate-Based Optimization in Process Systems

The Role of Surrogates in Data-Driven Optimization

Surrogate-based optimization, a class of model-based derivative-free methods, is particularly valuable when optimizing costly black-box functions [7] [1]. These scenarios arise when the objective function or constraints are determined by expensive experiments (e.g., in-vitro chemical experiments) or high-fidelity simulations (e.g., computational fluid dynamics) where derivatives are unavailable or unreliable [1]. The core idea is to construct a computationally efficient surrogate model (or meta-model) that approximates the expensive true function based on a limited set of evaluated data points [7] [1]. The optimizer then primarily works with this surrogate to navigate the design space efficiently.

Two prominent perspectives for developing these models are:

• Surrogate-Led Approach: The data-driven model takes precedence in guiding the optimization search.
• Mathematical Programming-Led Approach: The surrogate is tightly integrated into a traditional optimization framework, with the algorithm dictating the search trajectory [27].

A critical consideration in surrogate-based optimization is the verification problem—ensuring that the optimum found using the surrogate corresponds to the optimum of the underlying high-fidelity "truth" model [27].

Formulation for Black-Box Problems

In derivative-free optimization, the generic unconstrained problem is formulated as [1]:

[ \begin{align} \min_{\mathbf{x}} \quad & f(\mathbf{x}) \ \text{s.t.} \quad & \mathbf{x} \in \mathcal{X} \subseteq \mathbb{R}^{n_{x}} \end{align} ]

However, unlike in traditional optimization, there is no analytical expression for ( f(\mathbf{x}) ) to compute derivatives [1]. The algorithm must strategically explore the space, balancing the need to gather information about the function (exploration) with the goal of using existing information to find the optimum (exploitation) [1]. Termination is often based on a maximum number of function evaluations or runtime, as ensuring convergence to a true optimum is challenging when the function itself is unknown [1].

Table 2: Key Considerations for Surrogate-Based Optimization Formulation

Aspect	Consideration	Implication for Formulation
Function Evaluation Cost	Evaluations are computationally expensive or time-consuming [1].	The optimization algorithm must be sample-efficient. The number of function evaluations is a key performance metric.
Noise	Deterministic models can be corrupted by computational noise, making numerical derivatives unreliable [1].	Algorithms must be robust to noise. Smoothing or stochastic modeling techniques may be required.
Constraints	Constraints may also be black-box functions [7].	Constraint handling methods (e.g., penalty functions, feasible region modeling) must be integrated into the surrogate framework.
Dimensionality	Problems can be high-dimensional (e.g., reactor control) [7] [1].	The choice of surrogate model (e.g., Random Forests, Bayesian Optimization) must scale effectively with the number of variables.

Experimental Protocols and Implementation

A Protocol for Problem Formulation

• Define the Goal: Precisely state the engineering goal in qualitative terms. Example: "Minimize the energy consumption of a distillation column while maintaining product purity above 99.5%."
• Identify and Bound Decision Variables: List all parameters under the optimizer's control. Categorize as continuous or discrete. Establish realistic physical bounds for each variable (e.g., ( 300 \text{K} \leq T \leq 500 \text{K} )). Start with a minimal set to simplify initial problem solving [24].
• Formalize the Objective Function: Translate the qualitative goal into a single, quantifiable scalar function. For multiple objectives, employ techniques like Pareto optimization or weighted sum methods [28].
• Specify All Constraints: Document all equality and inequality constraints derived from physical limits, safety regulations, product specifications, and operational requirements [26] [24].
• Select the Modeling Paradigm: Determine if the problem can be modeled with analytical expressions (suitable for gradient-based optimizers [24]) or requires a black-box/surrogate-based approach due to expensive or noisy function evaluations [7] [1].
• Iterate and Refine: Begin with a simplified problem and achieve convergence. Subsequently, analyze the results, check for physical meaningfulness, and gradually increase the problem's complexity by adding variables or constraints, re-optimizing at each stage [24].

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Computational Tools for Surrogate-Based Optimization

Tool / Algorithm	Category	Primary Function	Typical Use Case
Bayesian Optimization (BO) [7] [1]	Surrogate-Based / Model-Based DFO	Uses probabilistic models to balance exploration and exploitation.	Global optimization of expensive black-box functions.
TuRBO [7]	Surrogate-Based / Model-Based DFO	A state-of-the-art BO method that uses trust regions.	High-dimensional, stochastic optimization problems.
CONLABy Linear Approximation) [7]	Surrogate-Based / Model-Based DFO	Constructs linear approximations for derivative-free optimization.	Low-dimensional constrained problems.
ENTMOOT [7]	Surrogate-Based / Model-Based DFO	Uses ensemble tree models (e.g., GBDT) as surrogates.	Problems where tree-based models provide high accuracy.
Particle Swarm Optimization [1]	Direct DFO / Metaheuristic	A population-based method inspired by social behavior.	Global search for non-convex or noisy problems.
Nelder-Mead Simplex [1]	Direct DFO	A pattern search method that operates on a simplex geometry.	Local optimization of low-dimensional problems without derivatives.
SB-218078	SB-218078, CAS:135897-06-2, MF:C24H15N3O3, MW:393.4 g/mol	Chemical Reagent	Bench Chemicals
SB-273005	SB-273005, CAS:205678-31-5, MF:C22H24F3N3O4, MW:451.4 g/mol	Chemical Reagent	Bench Chemicals

Formulating an optimization problem with well-defined decision variables, constraints, and objectives is a foundational step in applying surrogate-based techniques to process systems engineering. This formulation dictates the effectiveness and efficiency of the optimization algorithm, especially when dealing with costly black-box functions prevalent in chemical engineering and drug development. By adhering to structured protocols—starting simple, carefully selecting variables and constraints, and iteratively refining the problem—researchers can navigate complex design spaces to discover optimal, feasible, and meaningful solutions. The integration of robust surrogate modeling techniques ensures that these data-driven optimization strategies are both computationally tractable and scientifically sound, enabling advancements in automated control and decision-making for complex processes.

A Deep Dive into SBO Algorithms and Their Real-World Biomedical Applications

In the realm of process systems engineering, the optimization of complex systems is often hampered by computationally expensive simulations. Surrogate modeling, also known as metamodeling, has emerged as a powerful technique that uses simplified models to mimic the behavior of these complex, computationally intensive simulations [29]. By acting as efficient proxies, surrogate models enable faster evaluations and make large-scale optimization feasible across various engineering domains, including pharmaceutical manufacturing, materials design, and medical device development [15] [30] [29]. The fundamental premise of surrogate-based optimization is to replace the expensive "black-box" function evaluations with inexpensive approximations, thereby dramatically reducing computational costs while maintaining acceptable accuracy [1] [31].

The adoption of surrogate modeling is particularly valuable in process systems engineering where competing objectives need to be balanced, such as minimizing production costs while maximizing product purity in pharmaceutical manufacturing [15] [29]. These models provide engineers with the capability to perform extensive sensitivity analyses, explore design spaces thoroughly, and identify optimal trade-offs between conflicting objectives—tasks that would be prohibitively expensive using full-scale simulations [29]. As the pharmaceutical sector increasingly depends on advanced process modeling techniques to streamline drug development and manufacturing workflows, surrogate-based optimisation has emerged as a practical and efficient solution for driving substantial improvements in operational efficiency, cost reduction, and adherence to stringent product quality standards [15].

Taxonomy of Surrogate Modeling Techniques

Surrogate models can be broadly classified into several distinct categories based on their mathematical foundations, implementation complexity, and application domains. The taxonomy presented below encompasses the spectrum from traditional polynomial approaches to advanced machine learning techniques.

Table 1: Classification of Primary Surrogate Modeling Techniques

Model Type	Mathematical Foundation	Key Strengths	Common Applications
Polynomial Response Surfaces (PRS)	Low-order polynomial equations [29]	Simplicity, interpretability, low computational cost [29]	Stress-strain prediction, system dynamics approximation [29]
Kriging/Gaussian Process Regression (GPR)	Gaussian process theory, spatial correlation [29]	Uncertainty quantification, handles nonlinearity [32] [29]	Stent geometry optimization, structural mechanics [29]
Radial Basis Functions (RBF)	Basis functions dependent on distance from centers [31]	Good interpolation properties, handles irregular data [31]	High-dimensional expensive optimization [31]
Polynomial Chaos Expansion (PCE)	Orthonormal polynomial series [33]	Global sensitivity analysis, uncertainty quantification [33]	Atmospheric chemistry models, inverse modeling [33]
Artificial Neural Networks (ANN)	Layers of interconnected nodes inspired by biological brains [29] [33]	Captures complex nonlinear relationships, handles large datasets [29]	Fluid flow optimization, biological response prediction [29]
Support Vector Machines (SVM)	Statistical learning theory, kernel methods [30]	Effective with limited data, handles high-dimensional spaces [30]	Microstructural optimization of materials [30]

Traditional and Machine Learning-Based Surrogates

Beyond the fundamental classification, surrogate models can be further categorized as either traditional mathematical approximations or modern machine learning techniques. Traditional methods include Polynomial Response Surfaces, Kriging, and Polynomial Chaos Expansion, which are typically grounded in well-established mathematical principles and often provide greater interpretability [29] [33]. In contrast, machine learning-based surrogates such as Artificial Neural Networks and Support Vector Machines excel at capturing complex, nonlinear relationships in high-dimensional spaces but often require larger training datasets and offer less interpretability [29].

Another important distinction lies in their implementation strategies: static surrogates are constructed prior to the optimization process, often using simplified physics or relaxed internal tolerances, while dynamic surrogates are built and updated iteratively as the optimization progresses [31]. Research has also explored hybrid approaches, such as combining static surrogates as input for quadratic models within optimization algorithms like Mesh Adaptive Direct Search (MADS) [31].

Comparative Analysis of Surrogate Models

Understanding the relative strengths and limitations of different surrogate modeling techniques is crucial for selecting the appropriate approach for specific applications in process systems engineering.

Table 2: Performance Comparison of Surrogate Modeling Techniques

Model Type	Data Efficiency	Computational Cost	Handling Nonlinearity	Uncertainty Quantification	Interpretability
Polynomial Response Surfaces	High [29]	Low [29]	Low to moderate [29]	No	High [29]
Kriging/GPR	Medium [29]	Medium to high [29]	High [29]	Yes [32] [29]	Medium
Radial Basis Functions	Medium	Medium	Medium to high [31]	Limited	Medium
Polynomial Chaos Expansion	Medium [33]	Medium [33]	Medium to high [33]	Yes [33]	Medium to high
Artificial Neural Networks	Low (requires more data) [29]	High (training phase) [29]	Very high [29]	Limited	Low [29]
Support Vector Machines	Medium to high [30]	Medium to high	High [30]	Limited	Low to medium

Guidelines for Model Selection

Selecting the appropriate surrogate model depends on multiple factors, including the characteristics of the underlying process, available computational resources, and the specific objectives of the optimization study. For early-stage design exploration or problems with relatively smooth response surfaces, Polynomial Response Surfaces remain a practical choice due to their simplicity and low computational requirements [29]. When dealing with highly nonlinear systems with limited data and a need for uncertainty quantification, Kriging models are particularly advantageous [29]. For problems involving large datasets and complex, nonlinear relationships, Artificial Neural Networks often provide superior performance despite their higher computational demands and reduced interpretability [29].

In practical applications, researchers often employ ensemble approaches that combine multiple surrogate types to leverage their respective strengths [31]. Additionally, the choice of surrogate model may evolve throughout an optimization campaign, starting with simpler models for initial exploration and progressing to more sophisticated techniques as the region of interest becomes more defined.

Experimental Protocols and Implementation Frameworks

General Workflow for Surrogate-Assisted Optimization

The implementation of surrogate-based optimization follows a systematic workflow that integrates data generation, model training, and iterative refinement. The following diagram illustrates this generalized framework:

This workflow visualization captures the iterative nature of surrogate-based optimization, highlighting the critical stages of data generation, model training, and convergence checking before proceeding to final optimization.

Protocol 1: Surrogate Model Development for Pharmaceutical Process Optimization

Objective: To develop and validate surrogate models for optimizing Active Pharmaceutical Ingredient (API) manufacturing processes with multiple competing objectives (yield, purity, Process Mass Intensity) [15].

Materials and Software Requirements:

• Process Simulation Software: High-fidelity model of the API manufacturing process (e.g., dynamic system model) [15]
• Sampling Tool: Design of Experiments implementation with Latin Hypercube Sampling capability [32] [34]
• Surrogate Modeling Framework: Software with multiple surrogate modeling techniques (Gaussian Process Regression, Polynomial Chaos Expansion, Neural Networks) [32]
• Optimization Algorithms: Single- and multi-objective optimization algorithms compatible with surrogate models [15] [1]

Step-by-Step Procedure:

• Parameter Selection and Range Definition:

  • Identify critical process parameters through sensitivity analysis [34]
  • Define feasible ranges for each parameter based on operational constraints
  • For API manufacturing, key parameters may include temperature, pressure, reaction time, and catalyst concentration [15]

• Design of Experiments:

  • Generate sampling plan using Latin Hypercube Sampling to ensure space-filling properties [32] [31]
  • For 5-30 parameters, typically generate 100-10,000 sample points depending on complexity [34]
  • Balance sample size between computational cost and model accuracy requirements

• High-Fidelity Simulation:

  • Execute full process simulations at each sample point [15] [34]
  • Record key performance metrics (yield, purity, Process Mass Intensity) [15]
  • Validate simulation results against experimental data if available

• Surrogate Model Training:

  • Partition data into training (80%) and testing (20%) sets [34]
  • Train multiple surrogate types (PRS, Kriging, ANN, PCE) on training data
  • Optimize hyperparameters for each model type using cross-validation

• Model Validation:

  • Evaluate trained models on test dataset using metrics (R², RMSE, MAE) [34]
  • Select best-performing model based on accuracy and computational efficiency
  • For Kriging and PCE models, examine uncertainty estimates [32] [33]

• Implementation in Optimization Framework:

  • Integrate validated surrogate into optimization workflow [15]
  • Configure optimization algorithms (single- or multi-objective) [15] [1]
  • Establish convergence criteria based on objective function improvement or maximum iterations

Expected Outcomes: The protocol should yield validated surrogate models capable of accurately predicting API process performance metrics. Successful implementation typically achieves 1.5-3.6% improvement in yield while maintaining or improving purity standards, as demonstrated in pharmaceutical case studies [15].

Protocol 2: Microstructural Optimization of Structural Materials

Objective: To optimize microstructural features of structural materials (e.g., wrought aluminum alloys) for enhanced mechanical properties using surrogate modeling [30].

Materials and Software Requirements:

• 3D Image-Based Simulation: Finite element software capable of simulating mechanical behavior based on microstructural data [30]
• Microstructural Characterization Tools: Quantitative analysis of size, shape, and spatial distribution of microstructural features [30]
• Feature Selection Algorithm: For reducing dimensionality of design parameters (e.g., from 41 to 4 key parameters) [30]
• Support Vector Machine Framework: With infill sampling criterion for efficient optimization [30]

Step-by-Step Procedure:

• Microstructural Quantification:

  • Acquire 3D microstructural data using tomography techniques [30]
  • Quantify size, shape, and spatial distribution parameters of microstructural features
  • Initially identify 40+ potential design parameters describing microstructural characteristics [30]

• Parameter Space Coarsening:

  • Implement two-step coarsening process to reduce parameter dimensionality [30]
  • Apply feature importance analysis to identify most influential parameters
  • Finalize 3-5 critical parameters that dominantly affect mechanical performance

• Training Data Generation:

  • Perform 3D image-based numerical simulations for varied parameter combinations [30]
  • Define objective functions related to particle damage assessment [30]
  • Limit simulations to computationally feasible number (typically few hundred)

• SVM Surrogate Model Development:

  • Train Support Vector Machine models with infill sampling criterion [30]
  • Implement active learning strategy to focus simulations on promising regions
  • Validate model predictions against additional numerical simulations

• Microstructural Optimization:

  • Apply optimization algorithms to identify optimal microstructural parameters [30]
  • Balance competing objectives (strength, durability, damage resistance)
  • Verify optimal configurations through targeted validation simulations

Expected Outcomes: Identification of optimal microstructural characteristics (e.g., small, spherical particles with sparse dispersion perpendicular to loading direction) that enhance mechanical performance while suppressing internal stress concentrations [30].

Implementing surrogate-based optimization requires both computational tools and methodological frameworks. The following table outlines key components of the researcher's toolkit for successful surrogate modeling applications in process systems engineering.

Table 3: Essential Research Toolkit for Surrogate-Based Optimization

Tool Category	Specific Tools/Techniques	Function/Purpose	Application Context
Sampling Methods	Latin Hypercube Sampling (LHS) [32] [31]	Space-filling experimental designs for computer experiments [32] [31]	Initial training data generation [32]
Infill Criteria	Expected Improvement (EI) [31]	Balances exploration and exploitation during optimization [31]	Sequential sample selection [31]
Sensitivity Analysis	Polynomial Chaos Expansion (PCE) [33]	Global sensitivity analysis to identify influential parameters [33]	Parameter screening and reduction [30] [33]
Optimization Algorithms	Bayesian Optimization (BO) [1]	Efficient global optimization for expensive black-box functions [1]	Single and multi-objective optimization [15] [1]
Uncertainty Quantification	Gaussian Process Regression (Kriging) [32] [29]	Provides uncertainty estimates with predictions [32] [29]	Reliability-based design optimization [31]
Software Environments	COMSOL [32], MATLAB [34], Python/Keras [33]	Integrated platforms for simulation and surrogate modeling [32] [34] [33]	End-to-end implementation [32] [34] [33]

Applications in Process Systems Engineering and Biomedical Fields

Pharmaceutical Process Optimization

Surrogate-based optimization has demonstrated significant value in pharmaceutical manufacturing, where it enables simultaneous improvement of multiple critical quality attributes. Case studies show that unified surrogate optimization frameworks can achieve a 1.72% improvement in Yield and a 7.27% improvement in Process Mass Intensity in single-objective optimization, while multi-objective approaches deliver a 3.63% enhancement in Yield while maintaining high purity levels [15]. These improvements are particularly notable given the stringent regulatory requirements and complex multi-step processes characteristic of pharmaceutical manufacturing.

The application of surrogate modeling in quantitative systems pharmacology (QSP) has revolutionized virtual patient creation, where machine learning surrogates pre-screen parameter combinations to efficiently identify plausible virtual patients [34]. This approach addresses the challenge of computational expense in mechanistic QSP models, which traditionally required evaluating thousands of parameter combinations to find viable virtual patients. By using surrogates for pre-screening, researchers can focus full model simulations only on the most promising parameter sets, dramatically improving computational efficiency [34].

Materials Design and Medical Device Development

In materials science, surrogate modeling enables the optimization of microstructural features to enhance material performance. The integration of limited 3D image-based numerical simulations with microstructural quantification and optimization processes has proven effective for designing structural materials with superior properties [30]. This approach successfully handles complex design spaces with multiple parameters quantitatively expressing size, shape, and spatial distribution of microstructural features.

Medical device design represents another promising application area, where surrogate models help balance competing objectives such as minimizing device size while maximizing strength or ensuring durability without compromising biocompatibility [29]. The technique enables comprehensive design space exploration with significantly reduced computational cost compared to traditional finite element analysis or computational fluid dynamics simulations [29]. For stent development, for instance, sensitivity analysis using surrogate models can reveal how changes in strut thickness or material composition affect flexibility and restenosis risk, guiding refinements to enhance overall device performance [29].

The taxonomy of surrogate models encompasses a diverse spectrum of techniques, from traditional Polynomial Regression to advanced Deep Neural Networks, each with distinct characteristics, strengths, and optimal application domains. As demonstrated across pharmaceutical manufacturing, materials design, and medical device development, the strategic selection and implementation of appropriate surrogate modeling techniques can dramatically accelerate optimization cycles while maintaining necessary accuracy. The continued evolution of surrogate-based optimization methodologies, particularly through hybrid approaches and advanced machine learning techniques, promises to further enhance their utility in addressing complex challenges in process systems engineering and biomedical research.

Surrogate-based optimization techniques are indispensable in process systems engineering, particularly for optimizing complex, expensive-to-evaluate black-box functions where derivatives are unavailable or computational costs are prohibitive. These methods construct computationally efficient surrogate models to approximate the underlying system behavior, guiding the search for optimal conditions with remarkable sample efficiency. Within this domain, Bayesian Optimization (BO) and its advanced variants, such as Trust Region Bayesian Optimization (TuRBO), have emerged as powerful frameworks for global optimization in high-dimensional spaces. Their application is transformative for critical and costly domains like drug development and chemical process design, where physical experiments or high-fidelity simulations can be exceptionally time-consuming and expensive [35] [36].

This article provides detailed application notes and experimental protocols for deploying these advanced algorithms, with a specific focus on Bayesian Optimization and the highly scalable TuRBO method. The content is structured to equip researchers and scientists with practical methodologies for implementing these techniques in real-world process optimization and molecular discovery tasks.

Algorithmic Foundations and Comparative Analysis

Core Principles of Bayesian Optimization

Bayesian Optimization is a sequential design strategy for optimizing black-box functions. Its power derives from a probabilistic surrogate model, typically a Gaussian Process (GP), which provides a distribution over possible function values at any point in the search space. This surrogate is updated with each new function evaluation. An acquisition function, such as Expected Improvement (EI) or Upper Confidence Bound (UCB), leverages the surrogate's predictive mean and uncertainty to decide where to sample next, automatically balancing exploration (sampling in uncertain regions) and exploitation (sampling near promising known optima) [35]. This makes BO exceptionally data-efficient, a critical property when each function evaluation corresponds to a costly wet-lab experiment or a days-long simulation [36].

The TuRBO Advancement: Scalability via Local Modeling

A significant limitation of standard BO is its reliance on a global surrogate model, which can become inefficient in very high-dimensional spaces (dozens to hundreds of dimensions) or on functions with complex, localized structures. The Trust Region Bayesian Optimization (TuRBO) algorithm addresses this by combining BO with a local trust-region approach [37] [38].

Instead of a single global model, TuRBO maintains one or more local models within dynamic trust regions. The key innovation is that the size of each trust region is adapted based on performance: the region expands after a series of successful improvements and contracts after repeated failures. This allows TuRBO to focus computational resources on promising areas of the search space, avoiding over-exploration of barren regions. By leveraging local models, TuRBO achieves superior scalability and performance on high-dimensional problems, as demonstrated in its original publication where it outperformed global BO and other benchmarks on a challenging 20-dimensional Ackley function [37].

Quantitative Algorithm Comparison

The following table summarizes the core characteristics and optimal application domains for these algorithms, based on their documented performance.

Table 1: Comparative Analysis of Surrogate-Based Optimization Algorithms

Algorithm	Core Methodology	Key Strength	Typical Convergence Rate (Evaluations)	Ideal Problem Domain
Bayesian Optimization (BO)	Global Gaussian Process surrogate with acquisition function (e.g., EI, UCB).	High sample efficiency, strong theoretical guarantees.	100s - 1000s [36]	Low-to-moderate dimensional (≤20D) expensive black-box functions.
TuRBO (Trust Region BO)	Multiple local GP models within adaptive trust regions.	Scalability to high dimensions, robustness on complex landscapes.	~100 for 20D problems [37]	High-dimensional (20D+), non-convex functions with local structure.
MolDAIS (BO Variant)	Adaptive subspace identification within large molecular descriptor libraries using sparse priors.	Interpretability, automatic feature selection for molecular data.	<100 for >100k molecule search [35]	Molecular Property Optimization (MPO) with descriptor libraries.
ProfBO	Markov Decision Process (MDP) priors meta-learned from related task trajectories.	Extreme few-shot performance (<20 evaluations).	<20 evaluations [36]	Very high-cost evaluations with available data from related tasks.

Application Notes for Process Systems & Drug Development

Molecular Property Optimization (MPO) with MolDAIS

A prime application in modern drug development is the optimization of molecular structures for desired properties like binding affinity or solubility. The MolDAIS framework exemplifies a tailored BO approach for this domain. It operates on large libraries of precomputed molecular descriptors (e.g., atom counts, topological indices, quantum-chemical features) and uses a sparsity-inducing Sparse Axis-Aligned Subspace (SAAS) prior to automatically identify the most relevant descriptors during the optimization loop. This adaptive feature selection creates a parsimonious model that prevents overfitting and enhances sample efficiency, enabling the identification of near-optimal molecules from a pool of over 100,000 candidates in fewer than 100 property evaluations [35].

Table 2: Research Reagent Solutions for Molecular BO

Reagent / Software	Function in the Experimental Protocol
Molecular Descriptor Libraries (e.g., RDKit, Dragon)	Generates fixed-length feature vector representations of molecular structures for the surrogate model.
Gaussian Process Model with SAAS Prior	Serves as the probabilistic surrogate model; the SAAS prior enforces sparsity to focus on task-relevant features.
Acquisition Function (e.g., Expected Improvement)	Guides the selection of the next molecule to evaluate by balancing predicted performance and uncertainty.
Property Evaluation Software (e.g., DFT, molecular dynamics)	The "expensive black-box" that provides the property value (e.g., energy, solubility) for a given molecule.

Few-Shot Optimization with ProfBO for High-Cost Experiments

In scenarios like penicillin manufacturing or novel drug candidate screening, each evaluation can take weeks and cost millions of dollars. For these, a standard BO requiring hundreds of evaluations is impractical. The ProfBO algorithm is designed for this "few-shot" regime, finding global optima in fewer than 20 evaluations. Its core innovation is the use of Markov Decision Process (MDP) priors that are meta-learned from optimization trajectories of related source tasks (e.g., optimizing for different drug receptors or similar chemical processes). This allows ProfBO to leverage procedural knowledge of how to optimize effectively, not just data on the function's shape, leading to radically accelerated convergence on the new, costly target task [36].

Detailed Experimental Protocols

Protocol 1: Implementing TuRBO-1 for a High-Dimensional Problem

This protocol outlines the steps to apply the single trust-region TuRBO-1 algorithm, using a 20-dimensional Ackley function minimization as a benchmark [37].

Workflow Overview:

Step-by-Step Methodology:

• Problem Formulation:

  • Define the objective function f(x) to be minimized. Since BO typically maximizes, reformulate as max -f(x).
  • Normalize the input domain to [0, 1]^d for all d dimensions. The unnormalize function is used before true function evaluation.

• Algorithm Initialization:

  • Initialize the TurboState dataclass to track the trust region's history.
  • Key parameters: length=0.8 (initial trust region side length), length_min=0.57, length_max=1.6, success_tolerance=10, batch_size=4.
  • The failure_tolerance is automatically set to ceil(max(4.0 / batch_size, dim / batch_size)).

• Initial Design:

  • Generate n_init = 2 * dim initial evaluation points using a scrambled Sobol sequence for good space-filling properties.

• Main Optimization Loop:

  • a. Function Evaluation: Evaluate the expensive black-box function at the current batch of points.
  • b. State Update: Update the TurboState using the update_state function. The success/failure counters are updated based on whether a significant improvement (> 1e-3 * |best_value|) is found. The trust region length is expanded after success_tolerance consecutive successes and halved after failure_tolerance consecutive failures.
  • c. Model Fitting: Fit a local Gaussian Process model (e.g., with a Matern kernel) only to the data points within the current trust region.
  • d. Candidate Generation:
    • Thompson Sampling (Recommended): Generate a large number of candidate points within the scaled trust region. Use a perturbation mask to ensure diversity. Sample the next batch from these candidates using the GP's posterior.
    • qExpectedImprovement: Optimize the qEI acquisition function within the trust region bounds to select the next batch.
  • The loop continues until the evaluation budget is exhausted or the trust region shrinks below length_min, triggering a restart.

Protocol 2: Molecular Optimization with Adaptive Subspaces (MolDAIS)

This protocol is designed for data-efficient molecular discovery, leveraging the MolDAIS framework [35].

Workflow Overview:

Step-by-Step Methodology:

• Problem Setup:

  • Define the discrete molecular search space (e.g., a chemical library with >100,000 molecules).
  • Formally pose the Molecular Property Optimization (MPO) problem: m* = argmax F(m), where F is the expensive property function.

• Molecular Featurization:

  • Represent each molecule in the search space using a comprehensive library of molecular descriptors (e.g., using RDKit or Dragon). This results in a high-dimensional feature vector for each molecule.

• Adaptive Subspace BO Loop:

  • a. Sparse Model Training: Train a Gaussian Process surrogate model on the currently available (molecule, property) data. The critical component is the use of the SAAS prior, which promotes axis-aligned sparsity, effectively shutting out irrelevant descriptors.
  • b. Subspace Identification: The trained model automatically identifies a low-dimensional, property-relevant subspace of the original descriptor library. This subspace is iteratively refined as more data is collected.
  • c. Candidate Selection: Using an acquisition function like Expected Improvement, optimized within the adaptive subspace, select the most promising molecule to evaluate next.
  • d. Expensive Evaluation: Query the costly black-box function F(m)—a wet-lab experiment or high-fidelity simulation—to obtain the property value for the selected molecule.
  • e. Data Update & Iteration: Add the new (molecule, property) pair to the dataset and repeat steps a-d. The process typically requires fewer than 100 evaluations to identify near-optimal candidates.

Critical Analysis and Implementation Guidelines

Algorithm Selection Framework

Choosing the correct algorithm depends on the problem's constraints and data availability:

• For conventional, moderately expensive problems with a budget of hundreds of evaluations, standard BO with a GP surrogate is a robust default choice.
• For high-dimensional problems (20D+) or those with a complex, multi-modal landscape, TuRBO's local trust-region approach offers superior scalability and performance [37] [38].
• For molecular design and discovery, frameworks like MolDAIS that integrate domain-specific representations (descriptors) with adaptive feature selection are highly recommended for their interpretability and efficiency [35].
• For extremely high-cost problems where fewer than 20 evaluations are feasible and data from related tasks exists, ProfBO and its MDP priors represent the cutting edge for few-shot optimization [36].

Practical Considerations and Pitfalls

• Noisy Observations: The provided TuRBO-1 implementation is designed for noise-free or low-noise objectives. For problems with significant observation noise, the trust region center should be selected based on the GP posterior mean, not the best observed value.
• Computational Overhead: While BO reduces the number of expensive function evaluations, the computational cost of fitting the GP surrogate and optimizing the acquisition function can become non-trivial for very large initial datasets (>10,000 points). Sparse GPs and random restarts can help mitigate this.
• Meta-Learning Complexity: Methods like ProfBO require a set of related source tasks for meta-training the MDP priors. Their performance is contingent on the relevance and quality of these source tasks to the target problem.

The modeling of spatio-temporal dynamics is a cornerstone in understanding complex systems where processes evolve over both time and space, such as in fluid dynamics, epidemiology, and financial markets. Traditional approaches, predominantly based on the numerical approximation of differential equations, provide high fidelity but are often computationally prohibitive for many-query scenarios like design optimization, uncertainty quantification, or real-time control [39]. The emerging paradigm of surrogate-based optimization seeks to address this by replacing complex, expensive physics-based models with efficient, data-driven surrogates. Within this framework, deep learning architectures that synergistically combine Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) have demonstrated remarkable capability in learning the intrinsic dynamics of spatio-temporal systems directly from data [40] [41]. This document provides detailed application notes and protocols for leveraging these architectures to construct high-fidelity surrogates, with a particular emphasis on applications within process systems engineering and drug development.

Theoretical Foundations of Spatio-Temporal Deep Learning

Spatio-temporal data is characterized by its dual dependency on spatial and temporal dimensions, making its modeling uniquely challenging. The key is to capture both the spatial correlations (e.g., the structural relationship between different locations in a system) and the temporal dependencies (e.g., how the system state evolves over time) concurrently.

• Spatio-Temporal Graph Neural Networks (STGCNs): These models formulate the problem on graphs, where nodes represent spatial points (e.g., sensors in a traffic network, locations in a geographic area), and edges represent their spatial connectivity. STGCNs use complete convolutional structures to model the spatial dependencies, avoiding recurrent units to enable faster training and inference. This approach has been successfully applied to traffic forecasting, effectively capturing comprehensive spatio-temporal correlations [41].
• Spatio-Temporal Recurrent Neural Networks: Models like the Spatio-Temporal LSTM (ST-LSTM) are designed to memorize both spatial appearances and temporal variations in a unified memory pool. In architectures such as PredRNN, memory states are allowed to traverse both vertically across stacked RNN layers and horizontally through all RNN states, enabling a more effective learning of complex dynamics for tasks like video prediction [42].
• Latent Dynamics Networks (LDNets): This family of models introduces a novel approach by automatically discovering a compact, low-dimensional manifold in which the system dynamics evolve. An LDNet consists of two sub-networks: NN_dyn, which learns the dynamics of the latent variables via an ordinary differential equation, and NN_rec, which reconstructs the full output field from any point in space from these latent variables. This meshless approach avoids operations in high-dimensional space, is lightweight, and demonstrates superior accuracy and generalization, even in time-extrapolation regimes [39].

Table 1: Key Deep Learning Architectures for Spatio-Temporal Dynamics

Architecture	Core Principle	Key Advantage	Exemplary Application
Spatio-Temporal Graph CNNs (STGCN) [41]	Formulates problem on graphs; uses convolutional structures.	Faster training speed; captures spatial & temporal dependencies.	Traffic forecasting
Spatio-Temporal RNNs (e.g., PredRNN) [42]	Uses specialized LSTMs (ST-LSTM) with zigzag memory flow.	Unified memory for spatial and temporal features; state-of-the-art in prediction.	Video prediction
Latent Dynamics Networks (LDNet) [39]	Learns intrinsic dynamics in a low-dimensional latent space.	Meshless; lightweight; high accuracy in extrapolation.	General field prediction (e.g., fluid dynamics)
Spatio-Temporal Neural Networks [43]	Recurrent network with structured latent component and decoder.	Discovers spatial relations between time series.	Epidemiology, traffic prediction

Application Notes & Experimental Protocols

The integration of deep learning surrogates into process systems engineering, particularly for drug development, enables the optimization of complex, costly processes without repeatedly executing slow simulations or physical experiments.

Protocol 1: Building a Surrogate Model for a Reactor System Using LDNet

This protocol outlines the steps for creating a surrogate model to predict the spatio-temporal evolution of a concentration field within a chemical reactor, a common unit operation in pharmaceutical manufacturing.

1. Problem Definition and Data Generation

• Objective: Create a fast, data-driven surrogate for a computationally expensive Computational Fluid Dynamics (CFD) model of a tubular reactor.
• Inputs: Inlet flow rate u(t), inlet concentration C_in(t), and reactor geometry Ω.
• Output: The concentration field C(x, t) throughout the reactor over a specified time horizon.
• Data Generation: a. Define a range of realistic input signals for u(t) and C_in(t). b. Execute the high-fidelity CFD model for each input combination to generate the training dataset. c. For each simulation i, store the sequences of input signals u_i(Ï„) and the corresponding output fields C_i(ξ, τ) sampled at discrete spatial points ξ and times Ï„.

2. LDNet Model Configuration and Training

• Model Setup:
  • NN_dyn: A fully-connected neural network (FCNN) with 3 hidden layers (128 neurons each, tanh activation) that takes the latent state s(t) and input u(t) to compute ṡ(t).
  • NN_rec: An FCNN with 2 hidden layers (64 neurons each, tanh activation) that maps a query point x and the latent state s(t) to the predicted output C̃(x, t).
  • Latent dimension d_s: Start with 10 and tune as a hyperparameter.
• Training Loop: a. Initialization: Set latent state s(0) = 0. b. Time Integration: For each time step, integrate Equation (2) from the LDNet paper [39] using an ODE solver (e.g., Runge-Kutta 4th order) to obtain s(t). c. Reconstruction: Use NN_rec to predict the output field at the sensor locations ξ for each time t. d. Loss Calculation & Optimization: Minimize the mean squared error (MSE) between the predicted C̃(ξ, τ) and the true CFD data C(ξ, τ) using the Adam optimizer.

3. Surrogate Deployment and Optimization

• Once trained, the LDNet surrogate can replace the CFD model in a surrogate-based optimization loop.
• The optimization objective could be to find the inlet flow profile u(t) that maximizes product yield at the outlet, using an algorithm like Bayesian Optimization (BO) or Constrained Optimization by Quadratic Approximations (COBYQA) [7] [1].

Protocol 2: Multi-Channel Time Series Forecasting with STGCN for Supply Chain Responsiveness

This protocol applies to forecasting in networked systems, such as predicting inventory levels across a pharmaceutical supply chain to enhance responsiveness and sustainability [44].

1. Graph Construction and Data Preparation

• Graph Definition: Represent the supply chain network as a graph G = (V, E). Each node v_i ∈ V is a warehouse or distribution center. Edges E represent the transportation routes between them.
• Node Features: The primary feature for each node is its historical daily inventory level. Additional features can include incoming/outgoing orders and production rates.
• Data Collection: Gather time series data for all node features across the network for a period of 2-3 years.

2. STGCN Model Training

• Model Instantiation: Implement an STGCN model as described in [41]. It typically consists of:
  • Spatial Graph Convolutional Layers: To capture dependencies between neighboring warehouses (e.g., using Chebyshev polynomials or spectral methods).
  • Temporal 1D Convolutional Layers: To capture temporal patterns in the inventory data at each node.
• Training: The model is trained to predict the future H-step inventory levels for all nodes given the past P steps of historical data. The loss function is the MSE between predicted and actual inventory.

3. Scalability Assessment and Planning

• Use the trained STGCN model to forecast inventory under a wide range of simulated demand scenarios.
• Integrate these forecasts with a Mixed-Integer Linear Programming (MILP) model for network design and planning [44]. The α-shape method can be used to quantify the feasible solution region, indicating the supply chain's capacity to withstand demand variations.

Table 2: The Scientist's Toolkit: Key Research Reagents and Computational Materials

Item / Tool	Function in Spatio-Temporal Modeling	Exemplification in Protocol
High-Fidelity Simulator (CFD/PDE Solver)	Generates ground-truth data for training and validation.	High-fidelity CFD model of the chemical reactor.
Spatio-Temporal Graph	Defines the structural relationships (spatial topology) of the system.	Supply chain network graph with nodes (warehouses) and edges (routes).
Latent Dynamics Network (LDNet)	Learns and predicts system evolution in a low-dimensional, meshless manner.	Surrogate for the reactor's concentration field.
Spatio-Temporal Graph CNN (STGCN)	Forecasts future states in graph-structured time series data.	Forecasts inventory levels across the supply chain network.
Bayesian Optimization (BO)	Efficiently optimizes expensive black-box functions using a probabilistic surrogate.	Finds optimal inlet flow profile using the LDNet surrogate.
Multi-Objective Optimization	Solves problems with competing objectives (e.g., cost vs. sustainability).	Used with MILP for supply chain design considering cost and footprint [44].

Computational Methods and Surrogate-Based Optimization

The core of integrating these deep learning models into engineering workflows is surrogate-based optimization [7] [1]. The process involves:

• Initial Sampling: A limited number of data points are collected from the expensive "black-box" function (e.g., a high-fidelity simulation or a physical experiment).
• Surrogate Model Fitting: A spatio-temporal deep learning model (the surrogate) is trained on this initial data.
• Optimization Loop: a. An optimization algorithm (e.g., BO, COBYQA) queries the surrogate, which is cheap to evaluate, to propose a new candidate point that is likely to be optimal. b. The expensive black-box function is evaluated at this candidate point. c. The new data point is added to the training set, and the surrogate model is retrained or updated. d. The loop repeats until a convergence criterion is met, such as a maximum number of evaluations or minimal improvement.

This approach is particularly powerful for problems where the black-box function is deterministic but corrupted by computational noise, a common scenario in complex simulations [1].

Advanced Applications in Drug Development

The principles of spatio-temporal deep learning and surrogate optimization are transformative for the pharmaceutical industry, aligning with the push for a systems engineering approach [45].

• Enhanced Pharmaceutical Manufacturing: Transitioning from traditional batch processes to continuous pharmaceutical manufacturing is a key goal. STGCNs can model the complex, interconnected dynamics of a continuous processing line, predicting how perturbations in one unit (e.g., a reactor) affect downstream operations (e.g., a crystallizer). This enables the development of enhanced control systems that ensure consistent product quality and reduce waste [45].
• Predictive Pharmacokinetics/Pharmacodynamics (PK/PD): LDNet-style architectures can be used to build surrogates for physiologically-based PK models. These models predict drug concentration-time profiles in different tissues (a spatio-temporal field) based on administration route and patient parameters. A fast surrogate allows for rapid, personalized dosing optimization and reduced late-stage attrition in clinical development [45].

The fusion of convolutional and recurrent neural networks within architectures like STGCNs and LDNet provides a powerful, data-driven toolkit for modeling complex spatio-temporal dynamics. When embedded within a surrogate-based optimization framework, these models enable unprecedented efficiency in the design and control of process systems. For drug development, this means the potential for faster, more cost-effective, and more sustainable development of next-generation pharmaceuticals, from optimizing reactor conditions in active pharmaceutical ingredient (API) synthesis to planning robust and responsive supply chains. Future research will likely focus on improving the interpretability of these models, developing physics-informed versions to ensure predictions are physically plausible, and creating even more sample-efficient architectures to further reduce the data burden.

Sampling Strategies and Design of Experiments (DOE) for Efficient Data Collection

In the realm of process systems engineering, particularly within pharmaceutical development, optimization plays a pivotal role in enhancing cost-effectiveness, resource utilization, product quality, and process sustainability [15]. The rise of digitalization and complex chemical systems has led to the emergence of data-driven optimization (DDO) as a primary methodology, especially when data collection is only feasible through the evaluation of an expensive black-box function [1]. These functions may represent in-vitro chemical experiments with undetermined mechanisms, costly process reconfigurations, or in-silico simulations like computational fluid dynamics. The core challenge lies in the computational expense and potential noise of these evaluations, which makes even numerical derivatives difficult and unreliable. Surrogate-based optimisation emerges as a practical and efficient solution, where the optimization of a complex, expensive system is guided by a cheaper, approximate model built from data collected via strategic sampling and Design of Experiments (DOE) [15] [1].

Foundational Principles of Sampling and Experiment Design

The Role of Sampling in Model Generalization

Effective sampling is critical for developing models that generalize well to unseen process behavior. Traditional random sampling of event logs in predictive process monitoring can lead to Long Short-Term Memory (LSTM) models with a limited ability to generalize, primarily because the event logs often fail to capture the full spectrum of behavior permitted by the underlying processes [46]. To overcome this, innovative validation set sampling strategies, such as control-flow variant-based resampling, have been developed. These strategies ensure that the validation set used for hyperparameter tuning and early stopping is representative of the underlying process structure, not just common behavioral variants. This leads to notable enhancements in the generalization capabilities of trained models and a more accurate interpretation of the underlying process models [46].

Key Concepts in Experimental Design

The design of experiments is governed by principles that ensure reliability and validity. Key among these are the concepts of variables and controls. Variables are elements that change during an experiment, while controls are elements kept constant to ensure that any observed effects can be attributed to the manipulated variables [47]. Furthermore, writing effective scientific procedures that prioritize safety and reliability is paramount to ensuring experiments are repeatable and yield accurate results [47].

Sampling Strategies for Surrogate Model Development

The development of accurate surrogate models relies heavily on the strategic collection of data. The following sampling approaches are fundamental.

Variant-Based Resampling

Variant-based resampling is a strategy designed to improve the generalization of predictive models in processes with discrete, sequential events (e.g., business processes). It involves constructing training and validation sets based on the control-flow variants (unique pathways) present in an event log, rather than through simple random sampling [46]. This ensures that the model is exposed to and validated against a broader representation of the possible process behaviors during training.

• Application: This method is particularly valuable for next-event prediction models using LSTM networks. It combats the model's tendency to only learn frequently observed paths, thereby enhancing its understanding of the underlying process dynamics [46].
• Impact: Experiments have shown that incorporating variant-based resampling into the validation set positively impacts generalization scores and leads to better hyperparameter selection through early stopping [46].

Space-Filling Designs

For continuous parameter spaces, space-filling designs aim to spread sample points as uniformly as possible throughout the entire region of interest. This is crucial for initial surrogate model development when little is known about the system's response.

• Latin Hypercube Sampling (LHS): A statistical method for generating a near-random sample of parameter values from a multidimensional distribution. It ensures that the projection of the sample points onto each parameter is evenly distributed.
• Application: Ideal for the initial sampling phase to build a first-pass global surrogate model without prior knowledge of the system's optimal regions.

Adaptive Sampling

Once an initial surrogate model is built, adaptive sampling (or sequential design) strategies become highly efficient. These methods use the information from existing samples to decide where to sample next, focusing computational resources on areas of high interest, such as regions near the suspected optimum or areas of high model uncertainty.

• Bayesian Optimization (BO): A prominent model-based DFO method that uses a probabilistic surrogate model (often a Gaussian Process) to model the objective function. It employs an acquisition function to balance exploration (sampling in uncertain regions) and exploitation (sampling where the model predicts a good outcome) [1].
• Application: Extremely effective for optimizing costly black-box functions, as it aims to find the global optimum with a minimal number of function evaluations.

A Protocol for Surrogate-Based Optimization in Pharmaceutical Processes

This protocol outlines a systematic approach for applying surrogate-based optimization to an Active Pharmaceutical Ingredient (API) manufacturing process, adapting methodologies from the literature [15] [1].

Experimental Workflow

The following diagram illustrates the complete iterative workflow for surrogate-based optimization, from problem definition to the final implementation of the optimized conditions.

Detailed Experimental Protocol

Phase 1: Problem Formulation and DOE

• Define Objective: Clearly specify the objective function to be optimized (e.g., API yield, purity, Process Mass Intensity). Define any constraints (e.g., operating ranges, safety limits) [15].
• Identify Critical Process Parameters (CPPs): Using process knowledge, select the input variables (e.g., temperature, pressure, reaction time, catalyst concentration) that significantly influence the objective.
• Select and Execute DOE: Choose an appropriate sampling design (see Section 3). For initial screening, a space-filling design like Latin Hypercube Sampling is recommended. Specify the range for each CPP and generate the set of experimental runs.
• Run Experiments/Simulations: Execute the designed experiments using the high-fidelity model or physical system. Record the resulting performance metrics for each run.

Phase 2: Surrogate Model Development and Validation

• Data Collection: Assemble a dataset where each row corresponds to a DOE run, with columns for CPP inputs and the resulting objective function value(s).
• Model Building: Train one or more surrogate models (e.g., Gaussian Process, Radial Basis Functions, Ensemble Trees like ENTMOOT) on the collected dataset [1].
• Model Validation: Validate the surrogate model using a hold-out validation set or cross-validation. The model must accurately predict the objective function before proceeding. Key metrics include R² and Mean Squared Error. If performance is unsatisfactory, return to Phase 1 to collect more data in informative regions.

Phase 3: Optimization and Implementation

• Surrogate Optimization: Use an efficient optimizer (e.g., COBYLA, SNOBFIT, TuRBO) to find the set of CPPs that minimize or maximize the objective function on the validated surrogate model [1].
• Convergence Check: Determine if the optimization has converged to a solution. Criteria can include a minimal improvement over several iterations or a maximum number of iterations.
• Final Verification and Implementation: Run the high-fidelity model or a physical experiment at the proposed optimal conditions to verify the predicted performance. Once confirmed, implement the optimal conditions in the manufacturing process.

Quantitative Comparison of Sampling and Optimization Algorithms

Selecting the right combination of sampling strategy and optimization algorithm is critical for efficiency. The table below summarizes key approaches.

Table 1: Comparison of Sampling Strategies and Surrogate-Based Optimization Algorithms

Method Category	Specific Method / Algorithm	Key Characteristics	Best-Suited Application	Performance Notes
Initial Sampling	Latin Hypercube Sampling (LHS)	Space-filling, projective properties	Building initial global surrogate models	Provides a good baseline coverage of the parameter space [1].
Initial Sampling	Variant-Based Resampling	Ensures coverage of behavioral variants	Sequential process data (e.g., event logs)	Improves model generalization to unseen process behavior [46].
Adaptive Sampling / Optimization	Bayesian Optimization (BO)	Probabilistic model (Gaussian Process), balances exploration/exploitation	Costly black-box functions with low-to-moderate dimensions	Effective for global optimization with limited evaluations; TuRBO is a state-of-the-art variant for high-dimensional problems [1].
Adaptive Sampling / Optimization	Constrained Optimization by Linear Approximation (COBYLA)	Linear approximations, handles constraints	Low-dimensional, constrained problems	Robust for problems with few variables and known constraints [1].
Adaptive Sampling / Optimization	SNOBFIT (Stable Noisy Optimization by Branch and Fit)	Responds well to computational noise	Noisy, costly objective functions	Designed for stability in the presence of numerical or experimental noise [1].
Adaptive Sampling / Optimization	ENTMOOT (Ensemble Tree Model Optimization Tool)	Uses tree-based models (e.g., XGBoost)	Problems with structured, categorical inputs	Leverages the strengths of gradient-boosted trees for surrogate modeling [1].

Successful implementation of DOE and surrogate-based optimization requires both physical and computational resources.

Table 2: Key Research Reagent Solutions and Essential Materials

Item Name	Function / Role in the Framework
High-Fidelity Process Model	A detailed computational model (e.g., in Aspen Plus, gPROMS) or a well-instrumented lab-scale reactor that serves as the "ground truth" for evaluating sample points. It is the costly black-box function being approximated [1].
Design of Experiments (DOE) Software	Software tools (e.g., JMP, Modde, Python pyDOE2 library) used to generate efficient sampling plans like Latin Hypercube designs, guiding the initial data collection campaign.
Surrogate Modeling Library	Computational libraries (e.g., Scikit-learn, GPy, ENTMOOT) for building and training approximate models like Gaussian Processes or Random Forests on the collected data [1].
Derivative-Free Optimizer	Implementation of optimization algorithms (e.g., COBYLA, SNOBFIT, Bayesian Optimization frameworks) that can find the optimum by querying the surrogate model, without needing gradient information [1].
Performance Metrics Suite	A set of quantitative metrics (R², RMSE, Mean Absolute Error) and visualization tools to validate the accuracy and robustness of the developed surrogate model before proceeding to optimization.

The pharmaceutical industry increasingly relies on advanced process modelling to streamline drug development and manufacturing workflows [15]. Utilizing these models for optimization can drive substantial improvements in operational efficiency, cost reduction, and adherence to stringent product quality standards [15]. However, the complexity and high computational demands of such first-principles models often necessitate alternative approaches, with surrogate-based optimisation emerging as a practical and efficient solution [15] [9]. This case study details the application of a novel surrogate-based optimisation framework to an Active Pharmaceutical Ingredient (API) manufacturing process, framed within broader research on Process Systems Engineering (PSE) [48]. PSE is the scientific discipline of integrating scales and components describing the behaviour of a physicochemical system via mathematical modelling, data analytics, design, optimization, and control [48]. The findings demonstrate that surrogate models can effectively approximate complex behaviours, providing a practical approach to robust optimisation while navigating trade-offs between competing objectives such as yield, purity, and sustainability [15].

Methodologies

Surrogate-Based Optimization Framework

The implemented framework integrates multiple software tools into a unified system for employing surrogate-based methods to tackle challenges associated with optimising complex system models representing real-world API manufacturing [15] [9]. The framework supports both single- and multi-objective optimisation versions, focusing on improving key metrics such as yield, purity, and sustainability [15].

• Core Concept: A surrogate model, also known as a metamodel, is a simplified, computationally efficient model constructed to approximate the input-output relationships of a high-fidelity, computationally expensive model. This allows for rapid exploration of the design space and efficient optimization.
• Workflow Integration: The framework seamlessly integrates steps from data collection and surrogate model training to optimization and validation, ensuring robust results.

The following workflow diagram illustrates the logical relationships and sequential steps in the surrogate-based optimization process:

Experimental Protocol for API Process Optimization

This protocol provides a detailed methodology for applying the surrogate-based optimization framework to a dynamic system model of an API manufacturing process.

1. Definition of Optimization Objectives and Variables

• Primary Objectives: Maximize process yield; Maximize product purity; Minimize Process Mass Intensity (PMI) as a key sustainability metric [15].
• Key Process Variables: Identify Critical Process Parameters (CPPs) as independent variables. These typically include reaction temperature, reaction time, catalyst concentration, and feed flow rates [49].
• Defined Ranges: Establish feasible operating ranges for each CPP based on process knowledge and constraints.

2. Design of Experiments (DoE) and Data Generation

• DoE Selection: Employ a space-filling DoE technique (e.g., Latin Hypercube Sampling) to efficiently explore the defined design space of CPPs [15].
• High-Fidelity Simulation: Execute the complex, dynamic API manufacturing process model for each set of conditions specified by the DoE.
• Data Collection: Record the resulting values for the objectives (yield, purity, PMI) for each simulation run, creating a dataset for surrogate model training.

3. Surrogate Model Development and Validation

• Model Choice: Select appropriate surrogate model types, such as Polynomial Response Surface Models, Gaussian Process Regression (Kriging), or Artificial Neural Networks, based on the problem's non-linearity [15].
• Model Training: Use the dataset generated in Step 2 to train the surrogate models, establishing a mathematical relationship between CPPs and each objective.
• Model Validation: Validate the predictive accuracy of the trained surrogate models using a separate, hold-out test dataset or cross-validation. Metrics such as R² and Root Mean Square Error (RMSE) should meet pre-defined thresholds before proceeding.

4. Formulation and Execution of the Optimization Problem

• Single-Objective Optimization: Configure the solver to optimize one primary objective (e.g., maximize yield) while treating others as constraints (e.g., purity > 99.5%).
• Multi-Objective Optimization: Formulate a multi-objective problem to simultaneously optimize conflicting goals (e.g., maximize yield and minimize PMI). The solution is a set of non-dominated solutions known as the Pareto front [15].
• Optimization Algorithm: Use a suitable optimization algorithm (e.g., NLP solver for single-objective, genetic algorithms like NSGA-II for multi-objective) to find the optimum using the fast-evaluating surrogate models.

5. Validation of Optimal Conditions

• High-Fidelity Validation: Run the original high-fidelity process model at the optimal conditions identified by the surrogate-based optimization.
• Result Comparison: Compare the objectives predicted by the surrogate model with the results from the high-fidelity model validation to confirm performance.

Research Reagent and Material Solutions

The following table details key materials and computational tools essential for implementing the described surrogate-based optimization framework.

Table 1: Essential Research Reagents and Tools for Surrogate-Based Optimization

Item Name	Function/Application	Specification Notes
High-Fidelity Process Model	Serves as the "virtual process" to generate accurate data for surrogate model training. Represents the complex API manufacturing kinetics and transport phenomena [15].	Often built in environments like Aspen Plus, gPROMS, or custom-coded in Python/MATLAB.
Surrogate Modelling Software	Provides the computational engine for building and validating the approximation models from process data [15].	Open-source (e.g., scikit-learn, GPy) or commercial libraries (e.g., MATLAB's Curve Fitting Toolbox, JMP).
Optimization Solver	Numerical algorithms used to find the best set of process parameters that maximize or minimize the objective functions [50].	NLP solvers (e.g., IPOPT), MILP solvers (e.g., CPLEX, Gurobi), and multi-objective evolutionary algorithms (e.g., NSGA-II).
Process Mass Intensity (PMI) Calculator	A key sustainability metric, calculated as the total mass of materials used in the process divided by the mass of the final API [15].	Lower PMI values indicate a more efficient and environmentally friendly process.

Results and Discussion

Quantitative Optimization Results

The application of the surrogate-based optimization framework to the API manufacturing flowsheet yielded significant improvements across both single- and multi-objective scenarios. The quantitative outcomes are summarized in the table below.

Table 2: Summary of Optimization Results for API Manufacturing Process

Optimization Scenario	Key Objective	Baseline Performance	Optimized Performance	Percentage Improvement
Single-Objective	Yield	Baseline Value	Optimized Value	+1.72% [15]
Single-Objective	Process Mass Intensity (PMI)	Baseline Value	Optimized Value	+7.27% [15]
Multi-Objective	Yield (while maintaining high purity)	Baseline Value	Optimized Value	+3.63% [15]

The single-objective optimization successfully identified process conditions that led to a 1.72% increase in yield and a more substantial 7.27% improvement in Process Mass Intensity, underscoring the framework's capability to enhance both economic and sustainability metrics [15]. Notably, the multi-objective optimization strategy achieved an even greater yield improvement of 3.63% while maintaining high purity levels, demonstrating the power of this approach to navigate trade-offs between competing objectives effectively [15].

Multi-Objective Optimization and Pareto Front Analysis

A central outcome of the multi-objective optimization is the generation of a Pareto front. This front visualizes the trade-off between conflicting objectives, such as yield versus purity or yield versus PMI. The Pareto front is a set of solutions where improving one objective necessarily leads to the deterioration of another, meaning there is no single "best" solution but rather a range of optimal compromises [15].

The following diagram conceptualizes the trade-off relationship visualized by a Pareto front in such a multi-objective optimization:

This visualization allows researchers and drug development professionals to make informed decisions based on overarching project or business goals. For instance, a decision-maker might select a solution from the middle of the Pareto front, balancing a high yield with an acceptable purity level, rather than choosing the solution with the absolute maximum yield which might come with an unacceptably low purity [15]. The use of Pareto fronts is a cornerstone of modern Process Systems Engineering for managing such competing objectives in complex systems [48] [50].

This case study demonstrates that the novel surrogate-based optimisation framework presents a robust and efficient methodology for enhancing pharmaceutical process systems. By leveraging surrogate models to approximate complex, computationally expensive simulations, the framework enables significant improvements in critical performance indicators, including yield, purity, and sustainability metrics like Process Mass Intensity [15]. The ability to perform multi-objective optimization and visualize trade-offs via Pareto fronts provides invaluable insights for decision-makers in drug development, allowing for strategic compromises that align with broader project goals [15]. This approach aligns with the core tenets of Process Systems Engineering, which seeks to integrate components and scales of physicochemical systems through mathematical modelling and optimization [48]. The successful application documented herein underscores the potential of surrogate-based methods to contribute to more efficient, cost-effective, and sustainable pharmaceutical manufacturing.

The design of prosthetic devices presents a complex engineering challenge, characterized by costly evaluations, the need for personalization, and multi-objective design goals. This case study explores the application of surrogate-based optimization techniques to address these challenges, with a specific focus on the design of a prosthetic socket and a prosthetic foot. It details the experimental protocols and presents quantitative results that demonstrate how surrogate models, including Kriging and polynomial response surfaces, can drastically reduce computational expense while enabling effective design optimization. The findings highlight the potential of these data-driven methods to improve biomechanical outcomes, enhance user comfort, and accelerate the development of advanced medical devices.

The integration of advanced engineering methodologies into the design of prosthetic devices is crucial for improving the quality of life for individuals with disabilities. Traditional design processes often rely on iterative physical prototyping and clinician experience, which can be time-consuming, expensive, and suboptimal [51] [52]. Surrogate-based optimization offers a powerful alternative by using simplified mathematical models to emulate the behavior of complex, computationally expensive simulations (e.g., Finite Element Analysis - FEA) or physical experiments [1] [29]. This approach allows for the rapid exploration of a vast design space to identify optimal configurations that balance multiple, often competing, objectives such as maximizing comfort, minimizing tissue strain, and replicating natural biomechanics.

This case study is framed within a broader thesis on applying process systems engineering principles to medical device design. It demonstrates how surrogate models act as "digital twins" of the design process, enabling efficient optimization before committing to physical prototypes. The following sections present two detailed application notes: one on the design of a transtibial prosthetic socket and another on the optimization of a metamaterial-based prosthetic foot.

Application Note 1: Surrogate-Based Optimization of a Transtibial Prosthetic Socket

Background and Objective

The prosthetic socket is the critical interface between the residual limb and the prosthetic device. An ill-fitting socket can cause discomfort, pain, and deep tissue injury, often leading to device rejection [52]. The objective is to optimize socket design to minimize interfacial pressure and soft tissue strain, thereby ensuring comfort and safety for the user. The challenge lies in the extensive anatomical variability between patients and the computational cost of high-fidelity FEA, which hinders rapid, patient-specific design.

Surrogate Modeling and Optimization Protocol

This protocol outlines the development of a Kriging surrogate model to predict the biomechanical response of a residual limb to different socket designs, enabling rapid optimization [52].

• Step 1: Parameterization and Design of Experiments (DoE). Define the input variables (design parameters) and their bounds. A parametric model is created based on a baseline geometry from an MRI scan. A Latin Hypercube Sampling or similar space-filling DoE is recommended to generate a set of training points that efficiently cover the design space.
• Step 2: Finite Element Analysis (FEA). For each design point in the DoE, execute a static FEA. The model simulates socket donning and the application of a 400 N axial load (representative of standing). Key output metrics, such as peak interface pressure and sub-dermal soft tissue strain, are recorded.
• Step 3: Surrogate Model Construction. A Kriging (Gaussian Process Regression) surrogate model is fitted to the FEA data. This model learns the complex, nonlinear relationship between the input design parameters and the output biomechanical responses.
• Step 4: Design Optimization. An optimization algorithm (e.g., a genetic algorithm or COBYLA) is used on the Kriging model to find the input parameters that minimize the objective function (e.g., peak pressure). The key advantage is that evaluating the surrogate model is computationally cheap (milliseconds) compared to a full FEA (hours or days).
• Step 5: Model Validation. The optimal design predicted by the surrogate is validated by running a final, high-fidelity FEA. This confirms that the performance metrics fall within acceptable limits.

Table 1: Key Parameters for Socket Optimization Surrogate Model

Category	Parameter	Symbol	Lower Bound	Upper Bound	Description
Morphology	Residuum Length	v1	-1 σ (Short)	+1 σ (Long)	Principal component describing surgical amputation height [52]
	Residuum Profile	v2	-1 σ (Bulbous)	+1 σ (Conical)	Principal component describing limb shape [52]
	Bone Length	v3	-15%	+30%	Tibia length relative to residuum length [52]
	Soft Tissue Stiffness	v4	35 kPa	55 kPa	Elastic modulus of residuum soft tissue [52]
Socket Design	Proximal Press Fit	v5	-2%	+6%	Socket rectification at the proximal end [52]
	Mid Press Fit	v6	-2%	+6%	Socket rectification at the mid-section [52]
	Distal Press Fit	v7	-2%	+6%	Socket rectification at the distal end [52]

The application of this protocol led to a highly efficient framework for socket design. The Kriging surrogate model achieved real-time predictions (~1.6 ms per evaluation) with high accuracy—less than 4 kPa error in pressure prediction and less than 3% error in soft tissue strain prediction compared to the full FEA. This represents a computational expense reduction of six orders of magnitude, making it feasible for clinical implementation [52].

Table 2: Summary of Socket Optimization Model Performance

Metric	Performance	Implication
Computational Speed	1.6 ms per prediction	Enables real-time design exploration and optimization in a clinical setting [52]
Prediction Error (vs. FEA)	< 4 kPa (Pressure), < 3% (Strain)	High-fidelity predictions suitable for guiding design decisions [52]
Reduction in Computational Expense	Six orders of magnitude	Makes FEA-guided design practical by overcoming the barrier of long solver times [52]

Application Note 2: Optimization of a Metamaterial-Based Prosthetic Foot

Background and Objective

A primary goal in prosthetic foot design is to replicate the natural gait of able-bodied individuals, thereby reducing gait asymmetry and the risk of secondary complications like osteoarthritis [53]. This case study focuses on optimizing the geometric and material parameters of a prosthetic foot, including the use of auxetic metamaterials, to minimize the difference between its vertical Ground Reaction Force (vGRF) profile and that of a natural limb.

Surrogate Modeling and Optimization Protocol

This protocol leverages a finite element model of the gait cycle and a nature-inspired optimization algorithm to personalize the prosthetic foot design.

• Step 1: Geometric Parameterization and FEM Setup. Create a parametric model of the prosthetic foot. Key parameters include the offset of the outer edge (Z1), the thickness of the metatarsophalangeal joint (Z2), and the radius of the medial longitudinal arch (Z3) [53]. Develop a dynamic FEM that simulates the gait cycle and outputs the vGRF.
• Step 2: Experimental Data Collection. Collect experimental vGRF data from able-bodied individuals during walking using a force plate. This data serves as the target for the optimization process.
• Step 3: Optimization Loop. The Virus Optimization Algorithm (VOA) is used to drive the optimization [53]. The VOA generates a population of candidate designs (viruses). For each candidate, the FEM is executed to simulate gait and generate a vGRF profile.
• Step 4: Objective Function Evaluation. The objective function is calculated as the difference (e.g., root mean square error) between the vGRF generated by the prosthetic foot model and the target experimental vGRF data.
• Step 5: Algorithm Convergence. The VOA iteratively updates the population of designs based on the objective function, converging toward an optimal set of geometric parameters (Z1, Z2, Z3) that minimize the difference in vGRF.

Table 3: Key Parameters for Prosthetic Foot Optimization

Parameter Name	Symbol	Description	Role in Optimization
Keel Offset	Z1	Offset of the outer edge of the sketch relative to the keel geometry [53]	Influences overall structural stiffness and energy return during roll-over [53]
Joint Thickness	Z2	Thickness of the metatarsophalangeal joint component [53]	Controls flexibility and response at a critical joint during toe-off [53]
Arch Radius	Z3	Radius of the medial longitudinal arch of the foot [53]	Affects load distribution and shock absorption during mid-stance [53]

The Scientist's Toolkit: Essential Research Reagents and Materials

Table 4: Key Materials and Computational Tools for Prosthetic Device Optimization

Item / Solution	Function / Application in Research
Finite Element Analysis (FEA) Software	Creates high-fidelity simulations of physical stresses, strains, and pressures on the prosthetic device and residual limb; used to generate data for surrogate model training [52] [53].
Kriging Model	A powerful surrogate modeling technique that provides both predictions and uncertainty estimates; ideal for optimizing nonlinear systems like socket-limb interaction with limited data [52] [29].
Polynomial Response Surface (PRS)	A simpler surrogate model useful for early-stage design exploration and problems with smooth, low-nonlinearity responses [29].
Virus Optimization Algorithm (VOA)	A metaheuristic optimization algorithm used to efficiently explore complex design spaces, such as prosthetic foot geometry, where traditional gradient-based methods may struggle [53].
Auxetic Metamaterials	Materials with a negative Poisson's ratio that expand laterally when stretched. They offer enhanced energy absorption and impact resistance, and can be tailored to better mimic the mechanical behavior of biological tissues [53].
Statistical Shape Model (SSM)	Captures population-level anatomical variability (e.g., in residual limb morphology); enables the creation of patient-specific models and ensures robust design across a target population [52].
SB-743921 free base	SB-743921 free base, CAS:618430-39-0, MF:C31H33ClN2O3, MW:517.1 g/mol
SC-236	SC-236, CAS:170569-86-5, MF:C16H11ClF3N3O2S, MW:401.8 g/mol

Visualization of Workflows

Socket Design Optimization

The following diagram illustrates the integrated computational protocol for optimizing prosthetic socket design using surrogate modeling.

Prosthetic Foot Optimization

The following diagram outlines the workflow for optimizing a prosthetic foot's mechanical performance using the Virus Optimization Algorithm.

Overcoming Key Challenges: Data Scarcity, Model Fidelity, and Optimization Reliability

Addressing the Limited Data Problem with Hybrid and Transfer Learning Models

In process systems engineering, the development of high-fidelity models for optimization, control, and scale-up is often constrained by the limited availability of experimental or plant data. This "limited data problem" is particularly acute in complex chemical processes such as fluid catalytic cracking (FCC) and pharmaceutical manufacturing, where data acquisition is expensive, time-consuming, or technologically challenging. Surrogate-based optimization provides a powerful framework for addressing these challenges by constructing computationally efficient approximation models that mimic the behavior of expensive simulations or physical experiments [54] [7]. This application note details protocols for integrating hybrid modeling and transfer learning strategies to enhance surrogate models in data-scarce environments, specifically within the context of process systems research.

Core Methodological Frameworks

Hybrid Mechanistic-Data Driven Modeling

Hybrid modeling synergistically combines first-principles mechanistic knowledge with data-driven approaches, leveraging the strengths of both paradigms while mitigating their individual limitations.

• Data-Mechanism Complementary Dominance (DMCD) Framework: A novel conceptual framework that automatically adjusts the proportional weights of mechanistic knowledge and industrial datasets based on optimization and prediction range characteristics. This framework employs a pre-training strategy where industrial big data serves as the pre-trained dataset, while a massive dataset generated by the mechanism model functions as the real dataset, significantly improving model credibility and robustness [55].
• Integrated Hybrid Model Architecture: This architecture typically features parallel model components where a mechanistic model (e.g., molecular-level kinetic model) runs alongside a data-driven surrogate (e.g., deep neural network). The outputs are integrated to provide final predictions, maintaining physical significance while enhancing computational efficiency [56] [57].

Transfer Learning Strategies

Transfer learning enables knowledge transfer from data-rich source domains to data-scarce target domains, significantly reducing data requirements for new applications.

• TL-to-CL Strategy for Data Increment Scenarios: A hybrid transfer learning to continual learning strategy is particularly effective for systems where data accumulates gradually over time. This approach utilizes transfer learning when available data is limited, then transitions to continual learning once sufficient data has been accumulated, effectively leveraging newly added data while avoiding negative transfer [58].
• Property-Informed Transfer Learning: Specifically designed for chemical process scale-up, this strategy incorporates mechanistic equations for calculating bulk properties directly into the neural network architecture. This bridges the data structure gap between laboratory-scale molecular composition data and pilot/industrial-scale bulk property data, enabling effective knowledge transfer across scales [56].
• Deep Transfer Learning Network Architecture: For complex reaction systems, a specialized architecture using three residual multi-layer perceptrons (ResMLPs) simulates the computational procedure of mechanistic models. The Process-based ResMLP handles process conditions, the Molecule-based ResMLP captures compositional features, and an Integrated ResMLP predicts final product distribution, enabling targeted fine-tuning based on specific scale-up requirements [56].

Quantitative Performance Comparison

The table below summarizes documented performance improvements achieved through hybrid and transfer learning approaches in various process systems engineering applications.

Table 1: Performance Metrics of Hybrid and Transfer Learning Models

Application Domain	Modeling Approach	Key Performance Metrics	Reference
Fluid Catalytic Cracking (FCC) Process Optimization	Integrated Hybrid Modeling & Surrogate Optimization	• Product yield prediction error: < 4.84%• 0.10 wt% increase in LNG yield• 1.58 wt% increase in gasoline yield• 1.05 wt% increase in diesel yield• 3.67% increase in product revenues	[57]
Cross-Building Energy Prediction	TL-to-CL Strategy (Transform time = 4-week)	Prediction Improvement Ratio (PIR) of 0.4 ~ 0.9 compared to traditional LSTM	[58]
Biogas-to-Methanol Plant Emissions Assessment	Response Surface Methodology (Surrogate Modeling)	• Computational time reduction: Two orders of magnitude• Mean relative error: < 1%	[59]
Crude Oil Direct Cracking Optimization	Many-Objective Surrogate Optimization (MOEA/D)	Gasoline-oriented process used 29 tons less crude oil and generated 46.77 tons less CO2 per $1M GDP	[55]

Experimental Protocols

Protocol: Development of a Hybrid Surrogate Model for Process Optimization

This protocol outlines the systematic development of a hybrid surrogate model for optimizing industrial processes such as fluid catalytic cracking, integrating both mechanistic and data-driven components [55] [57].

• Step 1: Hybrid Data Collection

  • Plant Data Acquisition: Collect historical operational data from Distributed Control Systems (DCS), Manufacturing Execution Systems (MES), and Laboratory Information Management Systems (LIMS). Key variables typically include reactor temperatures, pressures, flow rates, and feedstock properties (e.g., density, carbon residue content) [57].
  • Mechanistic Data Generation: Use validated first-principles models (e.g., molecular-level kinetic models, computational fluid dynamics simulations) to generate supplementary data across a wide range of operating conditions, filling gaps in the experimental data space [55] [56].

• Step 2: Data Preprocessing & Feature Selection

  • Cleanse datasets by handling missing values, removing outliers, and addressing noise.
  • Normalize or standardize process variables to ensure numerical stability during model training.
  • Identify the most influential input variables through iterative communication with domain experts and statistical techniques (e.g., mutual information, principal component analysis) [57].

• Step 3: Multi-Task Learning Model Construction

  • Develop a neural network architecture with shared hidden layers and multiple output layers to simultaneously predict different product yields (e.g., gasoline, diesel, LNG).
  • Train the model on the combined plant and simulation dataset, ensuring the model learns both real-world operational patterns and mechanistic consistency [57].

• Step 4: Surrogate-Based Optimization

  • Embed the trained hybrid surrogate model as the objective function within a non-linear programming (NLP) optimization framework.
  • Define operational constraints based on physical limits and safety requirements.
  • Implement a multi-objective evolutionary algorithm (e.g., MOEA/D) to identify Pareto-optimal operating conditions that balance economic, environmental, and operational objectives [55] [57].

• Step 5: Validation and Implementation

  • Validate optimized operating conditions against holdout plant data or through rigorous mechanistic simulation.
  • Deploy the model for online optimization with a closed-loop implementation that continuously updates predictions based on real-time process data [57].

Protocol: Cross-Scale Process Modeling via Deep Transfer Learning

This protocol enables the adaptation of laboratory-scale kinetic models for pilot or industrial-scale prediction using deep transfer learning, effectively addressing data discrepancies across scales [56].

• Step 1: Source Model Development (Laboratory Scale)

  • Develop a high-fidelity molecular-level kinetic model calibrated with detailed laboratory-scale experimental data.
  • Use the mechanistic model to generate a comprehensive dataset of molecular conversions under varied compositions and conditions [56].

• Step 2: Laboratory-Scale Data-Driven Model Training

  • Design a specialized neural network architecture (e.g., using three ResMLP modules) that mirrors the structure of the mechanistic model.
  • Train the network on the data generated in Step 1 to create an accurate, fast-to-evaluate data-driven surrogate of the laboratory-scale process [56].

• Step 3: Target Data Preparation & Augmentation (Pilot/Industrial Scale)

  • Collect limited pilot-scale or industrial data, which typically consists of bulk properties rather than detailed molecular compositions.
  • Apply data augmentation techniques to expand the pilot-scale dataset, ensuring it adequately represents the operational space [56].

• Step 4: Property-Informed Transfer Learning

  • Incorporate mechanistic equations for calculating bulk properties directly into the neural network architecture (e.g., as a dedicated output layer).
  • Freeze layers in the network that encode fundamental reaction mechanisms (e.g., the Molecule-based ResMLP), which remain consistent across scales.
  • Fine-tune the remaining layers (e.g., Process-based ResMLP and Integrated ResMLP) using the augmented pilot-scale data to capture scale-specific transport phenomena and operational parameters [56].

• Step 5: Model Appraisal and Optimization

  • Assess the accuracy of the transferred model by comparing its predictions against held-out pilot-scale data.
  • Use the validated cross-scale model for multi-objective optimization of pilot-scale process conditions to maximize desired product yields and minimize energy consumption or environmental impact [56].

Workflow Visualization

The Scientist's Toolkit

Table 2: Essential Research Reagents and Computational Tools

Category/Item	Function in Research	Application Context
Surrogate Modeling Toolbox (SMT)	Python library providing a collection of surrogate modeling methods, sampling techniques, and benchmarking functions. Notably supports derivative-based modeling.	General-purpose surrogate model construction for optimization, design space exploration, and sensitivity analysis [54].
Multi-Objective Evolutionary Algorithm (MOEA/D)	Evolutionary algorithm for solving many-objective optimization problems by decomposing them into single-objective subproblems.	Optimization across competing objectives (economic, environmental, societal) in complex chemical processes [55].
Domain Adversarial Neural Network (DANN)	Neural architecture that learns domain-invariant features, facilitating effective knowledge transfer between source and target domains.	Implementing transfer learning strategies for cross-building energy prediction or cross-scale process modeling [58].
Residual Multi-Layer Perceptron (ResMLP)	Feedforward neural network employing skip connections to mitigate vanishing gradient problems, enabling deeper architectures.	Core component of deep transfer learning networks for complex reaction systems [56].
Conditional Tabular GAN (CTGAN)	Generative adversarial network designed to synthesize realistic tabular data, addressing data scarcity issues.	Data augmentation for creating high-quality synthetic datasets when real data is limited [60].
Response Surface Methodology (RSM)	Statistical and mathematical technique for empirical model building and optimization using polynomial functions.	Constructing computationally efficient surrogate models for plantwide emissions assessment and energy estimation [59].
Long Short-Term Memory (LSTM)	Type of recurrent neural network capable of learning long-term dependencies in sequential data.	Baseline model for time-series prediction tasks such as building energy consumption forecasting [58].
SC-560	SC-560, CAS:188817-13-2, MF:C17H12ClF3N2O, MW:352.7 g/mol	Chemical Reagent
SC-58125	SC-58125, CAS:162054-19-5, MF:C17H12F4N2O2S, MW:384.3 g/mol	Chemical Reagent

Infill criteria are decision-making rules that guide the selection of new evaluation points in surrogate-based optimization, directly managing the exploration-exploitation trade-off to locate global optima efficiently. In computationally expensive problems, such as process simulation or drug formulation development, each function evaluation may require hours or days of computational time or costly laboratory experiments. The core challenge lies in balancing two competing objectives: exploitation of promising regions identified by the surrogate model to refine solutions, and exploration of uncertain regions to avoid missing superior solutions. These techniques are particularly valuable in process systems engineering for optimizing complex, black-box systems where derivative information is unavailable and traditional optimization methods prove ineffective.

Theoretical Foundations of Infill Criteria

The Exploration-Exploitation Dilemma

The exploration-exploitation trade-off represents a fundamental framework in sequential decision-making under uncertainty. Exploitation involves selecting decisions that appear optimal given current knowledge, while exploration involves gathering new information that may lead to better long-term outcomes [61]. In surrogate-based optimization, this translates to a tension between sampling where the model predicts good performance versus sampling where model uncertainty is high. Effective infill criteria quantitatively balance this trade-off, enabling efficient convergence to global optima without becoming trapped in local solutions.

Mathematical Formulation

In surrogate-based optimization, we consider a minimization problem without loss of generality: $$min{\mathbf{x}} f(\mathbf{x}), \mathbf{x} \in \mathcal{X} \subseteq \mathbb{R}^{nx}$$ where $f(\mathbf{x})$ is an expensive black-box function. A surrogate model $\hat{f}(\mathbf{x})$ approximates the true function based on limited initial evaluations. The infill criterion $a(\mathbf{x})$ then determines the next evaluation point(s) by balancing the predicted objective $\hat{f}(\mathbf{x})$ and the uncertainty $\hat{s}(\mathbf{x})$ of the surrogate model [62].

Classification and Comparison of Infill Criteria

Infill criteria can be categorized based on their primary focus in the exploration-exploitation spectrum. The table below summarizes the fundamental characteristics of prominent criteria.

Table 1: Classification of Primary Infill Criteria

Criterion	Primary Focus	Key Mechanism	Advantages	Limitations
Minimize Predicted (MP)	Exploitation	Selects point with best predicted value [62]	Fast convergence	Prone to local optima
Expected Improvement (EI)	Balanced	Maximizes expected improvement over current best [62]	Theoretical optimality properties	Computational complexity in parallelization
Probability of Improvement (PI)	Balanced	Maximizes probability of improving current best [62]	Intuitive formulation	Less aggressive than EI
Upper Confidence Bound (LCB)	Balanced	Uses confidence interval bound [63]	Single tunable parameter	Performance sensitivity to parameter
Pseudo-EI	Balanced exploration	Uses influence function for parallel points [62]	Effective parallelization	Added complexity

Advanced and Hybrid Criteria

Recent research has developed sophisticated infill strategies that adaptively manage the exploration-exploitation balance:

• Stage-adaptive criteria: Change focus during optimization, initially prioritizing convergence followed by diversity maintenance [64]
• q-point Expected Improvement: Multi-point generalization of EI for parallel evaluation [62]
• Knowledge-guided criteria: Incorporate domain knowledge to focus search in physically meaningful regions [65]
• Multi-surrogate criteria: Combine global and local surrogates for different exploration-exploitation roles [66]

Quantitative Performance Comparison

The effectiveness of infill criteria varies significantly across problem types, dimensions, and computational budgets. The following table synthesizes performance observations from comparative studies.

Table 2: Comparative Performance of Infill Criteria Across Problem Types

Criterion	Low-Dim Unimodal	Low-Dim Multimodal	High-Dim Problems	Constrained Problems	Parallel Efficiency
MP	Excellent	Poor	Moderate	Good with constraints	Moderate
EI	Good	Excellent	Good	Good	Challenging
PI	Good	Good	Moderate	Moderate	Challenging
DMP	Good	Excellent	Good	Good	Excellent
PEI	Good	Excellent	Good	Good	Excellent

Studies indicate that criteria incorporating both prediction mean and uncertainty (e.g., EI, PEI) generally outperform purely exploitative methods on multimodal problems, with DMP and PEI showing particular efficiency and robustness across diverse problem types [62]. On high-dimensional reactor control problems, adaptive Bayesian optimization methods including TuRBO have demonstrated superior performance [1].

Implementation Protocols

Standard EGO Implementation

The Efficient Global Optimization (EGO) algorithm provides the foundational framework for implementing infill criteria [62].

Protocol 1: Basic EGO with Expected Improvement

• Initial Design: Generate initial sample points $S = {x^1, x^2, ..., x^n}^T$ using space-filling design (e.g., Latin Hypercube Sampling)
• Function Evaluation: Compute expensive function values $y = {y^1(x), y^2(x), ..., y^n(x)}^T$
• Surrogate Modeling: Construct Kriging model providing both prediction $\hat{y}(x)$ and uncertainty $\hat{s}(x)$
• Infill Optimization: Find $x^* = argmax(EI(x))$ where $EI(x) = E[max(y_{min} - y(x), 0)]$
• Model Update: Evaluate $f(x^*)$, add to training data, and update surrogate
• Termination Check: Repeat steps 4-5 until evaluation budget exhausted or convergence achieved

Parallel Infill Sampling Protocol

For parallel computing environments, the pseudo-EI criterion enables efficient simultaneous evaluation of multiple points [62].

Protocol 2: Parallel Infill with Pseudo-EI Criterion

• Initialization: Complete steps 1-3 of Protocol 1
• Candidate Selection:
  • Identify first candidate $x1^* = argmax(EI(x))$
  • Calculate influence function to approximate model update without actual evaluation
  • Identify subsequent candidates $x{2...q}^*$ using pseudo-EI criterion with reduced EI near previous candidates
• Parallel Evaluation: Simultaneously evaluate all q candidate points
• Batch Update: Add all new points to training data and update surrogate model
• Termination: Repeat until budget exhaustion

The adaptive distance function variant introduces a threshold mechanism to prevent candidate clustering and maintain global search capability [62].

Multi-Surrogate Framework with Global Exploration and Local Exploitation

Advanced frameworks combine different surrogate types to specialize in exploration versus exploitation [66].

Protocol 3: Global-Local Surrogate Framework

• Global Surrogate Construction: Build scalable Gaussian Process (GP) model with hyperparameter sharing for computational efficiency
• Global Exploration: Use GP prediction mean and uncertainty to identify promising regions
• Local Surrogate Construction: Build Radial Basis Function Network (RBFN) focused on promising regions identified by global surrogate
• Local Exploitation: Optimize within local search region ensuring bilateral coverage around current optimum
• Selective Evaluation: Evaluate only the local optimum with expensive true function
• Adaptive Region Adjustment: Dynamically adjust local search region based on solution location (especially boundary solutions)

This approach addresses limitations of using single surrogates and memetic methods under limited evaluation budgets [66].

Stage-Adaptive Infill for Multimodal Problems

For expensive multimodal multi-objective optimization, a stage-adaptive approach effectively balances convergence and diversity [64].

Protocol 4: Two-Stage Adaptive Infill

• Stage 1 - Convergence Focus:

  • Employ convergence-first infill sampling criterion
  • Use multiple surrogates (XGBoost for objectives, Self-Organizing Map for PS topologies)
  • Focus on driving population toward global Pareto front

• Stage Transition:

  • Trigger based on convergence metrics or evaluation count
  • Typically occurs after 30-40% of evaluation budget

• Stage 2 - Diversity Focus:

  • Implement indicator-based infill criterion using SOM neuron weights as reference vectors
  • Apply diversity-based criterion for objective space diversity
  • Maintain auxiliary population for decision-space diversity

This protocol specifically addresses the challenge of capturing multiple Pareto sets in expensive multimodal multi-objective problems [64].

Visualization of Workflows

Infill Criterion Implementation Workflow

The Researcher's Computational Toolkit

Table 3: Essential Computational Methods for Surrogate-Based Optimization

Method Category	Specific Techniques	Primary Application	Key References
Surrogate Models	Kriging (GP), RBFN, XGBoost, Ensemble Methods	Function approximation	[66] [62] [64]
Global Optimization	Bayesian Optimization, TuRBO, SNOBFIT	Black-box optimization	[7] [1]
Exploration-Exploitation	EI, UCB, Thompson Sampling, Adaptive Criteria	Infill decision making	[67] [62] [63]
Parallelization	Constant Liar, Pseudo-EI, KB	High-performance computing	[62]
Multi-objective	Stage-adaptive, SOM, Speciation	Multimodal problems	[64]
SEA0400	SEA0400|Na+/Ca2+ Exchanger (NCX) Inhibitor|CAS 223104-29-8		Bench Chemicals

Application in Process Systems Engineering

In process systems engineering, infill criteria enable optimization of expensive simulations including computational fluid dynamics, reactor design, and separation processes. Specific applications include:

• Chemical reactor optimization: Balancing conversion, selectivity, and energy consumption under uncertainty
• Drug development: Formulation optimization with expensive experimental verification
• Energy systems: Process integration with multiple conflicting objectives
• Materials design: Composition optimization with costly characterization

The hybrid analytical surrogate approach combining Bayesian symbolic regression with mechanistic knowledge has demonstrated particular effectiveness in process flowsheet optimization, finding better solutions than pure black-box approaches [65].

Ensuring Model Reliability and Robustness in Offline Black-Box Optimization

Offline black-box optimization is a critical paradigm for numerous science and engineering applications, including drug development and chemical process engineering, where evaluating candidate designs involves expensive, time-consuming physical experiments or computational simulations [68]. In this context, researchers must rely on a fixed, historical dataset to find optimal inputs without the ability to perform new evaluations of the true objective function. The prototypical approach involves learning a surrogate model from the training data to predict objective function values for unknown inputs [68]. However, these surrogate models are often only reliable within a constrained neighborhood of the offline data and can be highly erroneous outside this region, leading to significant performance gaps between the optima of the surrogate model and the true objective function [68]. This application note establishes comprehensive protocols for ensuring model reliability and robustness in offline black-box optimization, framed within the broader context of surrogate-based optimization techniques for process systems engineering research.

Foundational Concepts and Theoretical Framework

The Offline Black-Box Optimization Problem

Formally, the offline black-box optimization problem can be defined as follows [68]: Let 𝔛 be an input space where each 𝐱 ∈ 𝔛 is a candidate input (e.g., a molecular structure or process parameter configuration). Let 𝑔:𝔛↦ℜ be an unknown, expensive real-valued objective function that can evaluate any given input 𝐱 to produce output 𝑧 = 𝑔(𝐱). The goal is to find an optimal input or design 𝐱∗ that maximizes this objective:

𝐱∗ ≜ argmax┬(𝐱∈𝔛)𝑔(𝐱)

We are provided with a fixed dataset of 𝑛 input-output pairs 𝔇 = {(𝐱₁, 𝑧₁), (𝐱₂, 𝑧₂), ⋯, (𝐱ₙ, 𝑧ₙ)} where 𝑧ᵢ = 𝑔(𝐱ᵢ), with no access to the target objective function 𝑔 beyond this offline dataset.

Quantifying the Performance Gap

A fundamental theoretical advance in understanding model reliability comes from recent work on gradient matching, which provides a provable bound on the optimization performance gap [68]. This framework characterizes the offline optimization performance of gradient-based search guided by a surrogate model by bounding the performance gap between the optima of the target function and trained surrogate as a function of how well the surrogate matches the latent gradient field of the target function on the offline training data.

The derived bound demonstrates that the worst-case performance of an optimizer following the surrogate gradient is bounded by the gradient gap between the surrogate and target function, and that this bound is tight up to a constant with a sufficiently small learning rate [68]. This theoretical insight directly informs the practical protocol of gradient matching for developing reliable surrogate models.

Application Notes: Algorithms and Implementation

Surrogate-Based Optimization Techniques

Various surrogate-based optimization algorithms have demonstrated effectiveness in process systems engineering applications [7] [1] [69]. These can be broadly categorized as follows:

Table 1: Surrogate-Based Optimization Algorithms for Process Systems Engineering

Algorithm Category	Representative Methods	Key Characteristics	Applicable Scenarios
Bayesian Optimization	TuRBO, Standard BO	Probabilistic models, uncertainty quantification	High-dimensional problems, global optimization
Quadratic Approximation	COBYQA	Local quadratic models	Smooth objective functions
Tree-Based Methods	ENTMOOT	Decision trees as surrogates	Mixed-variable problems, interpretability
Radial Basis Functions	DYCORS, SRBFStrategy	Nonlinear function approximation	Computationally expensive simulations
Direct Search & Approximation	COBYLA, SNOBFIT	Linear approximation, branch-and-fit	Constrained optimization, noisy objectives

MATCH-OPT: A Gradient Matching Approach

Inspired by the theoretical framework linking gradient accuracy to optimization performance, the MATCH-OPT algorithm provides a principled approach to creating effective surrogate models for offline optimization [68]. This method is model-agnostic and allows approximation of the gradient field underlying the offline training data using a parametric surrogate.

The algorithm operates by explicitly training surrogate models to match the latent gradient field of the target function, which directly minimizes the optimization risk when following the surrogate's gradient toward the goal of finding the maximum of the target objective function [68]. Experimental results demonstrate that MATCH-OPT consistently shows improved optimization performance over existing baselines and produces high-quality solutions with gradient search from diverse starting points.

Experimental Protocols

Protocol 1: Gradient Matching Implementation

Objective: Implement and validate the MATCH-OPT gradient matching approach for offline black-box optimization.

Materials and Dataset Preparation:

• Collect or access an existing dataset of input-output pairs 𝔇 = {(𝐱ᵢ, 𝑧ᵢ)} where 𝑧ᵢ = 𝑔(𝐱ᵢ)
• Partition data into training (80%) and validation (20%) sets
• Normalize input features to zero mean and unit variance
• Standardize objective function values if necessary

Procedural Steps:

• Surrogate Model Selection: Choose an appropriate model architecture (e.g., neural network, Gaussian process) based on data size and complexity [68].

• Gradient Matching Training:

  • Define a loss function that combines prediction error and gradient matching terms
  • Incorporate gradient matching even when true gradients are unavailable by using:
    • Finite difference approximations between nearby data points
    • Regularization terms that encourage gradient smoothness
    • Physical constraints if available from domain knowledge

• Optimization Loop:

  • Initialize starting points from diverse regions of the input space
  • Perform gradient-based optimization on the trained surrogate
  • Implement trust region methods to constrain steps to regions where the surrogate is reliable

• Validation and Performance Assessment:

  • Compare found optima against held-out test points
  • Evaluate performance gap using the theoretical bounds
  • Assess robustness through multiple runs with different initializations

Expected Outcomes: A surrogate model that maintains reliable performance even outside the immediate neighborhood of the training data, with demonstrably smaller optimization performance gaps compared to standard regression approaches.

Protocol 2: Multi-Objective Optimization with Uncertainty Propagation

Objective: Implement surrogate-assisted optimization balancing multiple objectives under uncertainty, following the approach applied to γ-valerolactone (GVL) production [70].

Materials and Dataset Preparation:

• Kinetic model or historical data for the process of interest
• Latin Hypercube Sampling design for parameter uncertainty
• Computational resources for parallel simulation runs

Procedural Steps:

• Surrogate Model Development:

  • Generate training data using design of experiments
  • Develop surrogate models relating input variables to multiple outputs (e.g., production rate, risk index)
  • Validate surrogate accuracy against detailed physics-based models

• Uncertainty Propagation:

  • Employ Latin Hypercube Sampling to assess parameter uncertainty
  • Propagate uncertainties through the surrogate models
  • Quantify output distributions for key performance indicators

• Multi-Objective Optimization:

  • Formulate objective function balancing competing goals (e.g., maximize production rate, minimize thermal risk)
  • Implement multi-objective optimization algorithms (e.g., NSGA-II)
  • Generate Pareto fronts illustrating trade-offs between objectives

• Decision Support:

  • Identify optimal operating conditions under deterministic and uncertain scenarios
  • Validate critical points with high-fidelity models where feasible

Expected Outcomes: A set of Pareto-optimal solutions demonstrating the trade-offs between performance and safety, with quantified uncertainty bounds enabling robust decision-making.

Visualization Framework

Workflow for Reliable Offline Black-Box Optimization

The following diagram illustrates the comprehensive workflow for implementing reliable offline black-box optimization, integrating the key components discussed in this document:

Workflow for Reliable Offline Optimization

Gradient Matching Conceptual Architecture

The following diagram illustrates the conceptual architecture of the gradient matching approach, which is fundamental to ensuring model reliability in offline black-box optimization:

Gradient Matching Architecture

The Scientist's Toolkit

Research Reagent Solutions

Table 2: Essential Computational Tools and Materials for Offline Black-Box Optimization

Tool/Material	Function/Purpose	Implementation Examples
Surrogate Model Architectures	Function approximation from limited data	Gaussian Processes, Neural Networks, Decision Trees (ENTMOOT) [7] [1]
Gradient Matching Framework	Align surrogate gradients with target function	MATCH-OPT algorithm [68]
Uncertainty Quantification Tools	Propagate parameter uncertainties	Latin Hypercube Sampling, Monte Carlo Methods [70]
Multi-Objective Optimization Algorithms	Balance competing performance criteria	NSGA-II, Pareto front generation [70]
Benchmarking Suites	Validate algorithm performance	Design-bench benchmark [68], custom test functions
Theoretical Performance Bounds	Quantify optimization performance gaps	Gradient discrepancy measures [68]

Ensuring model reliability and robustness in offline black-box optimization requires a multifaceted approach combining theoretical foundations, algorithmic innovations, and rigorous validation protocols. The gradient matching framework provides a principled foundation for developing reliable surrogate models, while multi-objective optimization under uncertainty addresses practical engineering constraints. By implementing the protocols and methodologies outlined in this document, researchers and drug development professionals can enhance the reliability of their optimization outcomes in applications ranging from chemical process optimization to pharmaceutical development. The integration of theoretical performance bounds with practical algorithmic strategies creates a robust foundation for applying surrogate-based optimization techniques to challenging real-world problems where experimental data is limited and costly to obtain.

{Abstract} The gradient matching approach is an efficient surrogate-based optimization technique that circumvents the computationally expensive, repeated numerical integration of coupled ordinary differential equations (ODE)s. By matching the gradients of a data interpolant against those predicted by a mechanistic ODE model, it enables rapid and reliable parameter inference in complex dynamic systems. This Application Note details the theoretical framework, provides validated experimental protocols, and outlines essential computational reagents for applying gradient matching in process systems engineering and drug development research.

In systems biology and process engineering, the dynamics of complex networks are often modeled using systems of ODEs. A typical form for a species concentration ( xi(t) ) is: [ \frac{dxi(t)}{dt} = gi(\mathbf{x}(t), \boldsymbol{\rho}i, t) - \deltai xi(t) ] where ( \boldsymbol{\rho}i ) is a vector of kinetic parameters and ( \deltai ) is a decay rate [71] [72]. Conventional parameter inference methods require numerically solving these ODEs thousands of times, which is computationally prohibitive for large-scale systems [71].

The gradient matching approach provides a compelling alternative. Its core principle is to avoid explicit ODE integration by performing a two-step process: first, a smooth interpolant is fitted to the time-series data; second, the parameters of the ODE model are optimized by minimizing the difference between the slope of the interpolant and the time derivative predicted by the ODE model [71] [72]. This method effectively profiles over unknown initial conditions and dramatically reduces computational cost, making it particularly suitable for optimizing complex processes in pharmaceutical development and chemical engineering.

Comparative Analysis of Gradient Matching Methodologies

Various computational implementations of the gradient matching paradigm have been developed. The table below provides a structured comparison of the most prominent methods, highlighting their core algorithms, interpolation techniques, and key characteristics relevant for application in process systems.

Table 1: Comparative Analysis of Gradient Matching Methodologies

Method Name	Core Algorithm / Interpolation	Key Characteristics	Inference of Mismatch Parameter γ	Best-Suited Applications
GPM & AGM [71] [72]	Gaussian Process (GP)	Non-parametric Bayesian framework; infers all hyperparameters consistently.	Yes (Inferred as part of the model)	Complex, noisy biological systems; problems with unknown smoothness.
Two-Stage GM [71]	B-splines / RKHS	Simpler two-step process; inference quality highly dependent on initial interpolant.	No	Preliminary analysis; systems with high-quality, dense data.
RGM [71] [72]	B-splines	Hierarchical regularization; interpolants are regularized by the ODEs themselves.	No (Heuristic tuning)	Systems with well-characterized ODE structures.
PT-GM [71]	Gaussian Process (GP)	Uses parallel tempering to handle local optima; robust convergence.	No (Tempered)	Problems with complex, multi-modal likelihood surfaces.

The Adaptive Gradient Matching (AGM) with Gaussian Processes, as proposed by Dondelinger et al., is often the preferred method for complex systems. It defines the system's signals by the ODEs ( \dot{x}i = fi(\mathbf{X}, \boldsymbol{\theta}_i, t) ) and uses a Gaussian process prior ( p(\mathbf{X} | \boldsymbol{\phi}) ) for the latent variable ( \mathbf{X} ) [71] [72]. The key to its success is the co-inference of a parameter ( \gamma ) that controls the trade-off between fidelity to the data and fidelity to the ODE model, which prevents the inference from converging to poor local optima [71].

Application Protocols for Process Systems

This section provides detailed, step-by-step protocols for implementing the gradient matching approach, with a focus on parameter inference in dynamic models relevant to bioprocessing and drug development.

Protocol 1: Parameter Inference via Adaptive Gradient Matching

Purpose: To reliably estimate kinetic parameters ( \boldsymbol{\theta} ) in a system of ODEs from noisy, sparse time-course data.

Experimental Workflow:

Procedure:

• Data Preparation: Collect experimental or process time-course data ( \mathbf{Y} ). The data should be formatted into a matrix where rows represent time points and columns represent different measured species (e.g., metabolite concentrations, protein levels).
• Model Specification: Formulate the system of ODEs ( \dot{x}i = fi(\mathbf{X}, \boldsymbol{\theta}_i, t) ) that describes the underlying process dynamics. This constitutes the mechanistic "surrogate" model.
• Prior Selection: Place a Gaussian process prior on the latent states ( \mathbf{X} ), ( p(\mathbf{X} | \boldsymbol{\phi}) ), where ( \boldsymbol{\phi} ) are the GP hyperparameters (e.g., length-scale, amplitude) [71].
• Parameter Initialization: Provide initial guesses for the ODE parameters ( \boldsymbol{\theta} ), GP hyperparameters ( \boldsymbol{\phi} ), and the gradient mismatch parameter ( \gamma ).
• Posterior Sampling: Use Markov Chain Monte Carlo (MCMC) sampling (e.g., Hamiltonian Monte Carlo) to draw samples from the full posterior distribution: [ P(\boldsymbol{\theta}, \boldsymbol{\phi}, \gamma | \mathbf{Y}) \propto P(\mathbf{Y} | \mathbf{X}, \boldsymbol{\phi}) P(\mathbf{X} | \boldsymbol{\phi}) P(\boldsymbol{\theta}) P(\boldsymbol{\phi}) P(\gamma) ] The key step is comparing the GP gradient ( \dot{\mathbf{X}} ) to the ODE-predicted gradient ( \mathbf{f}(\mathbf{X}, \boldsymbol{\theta}, t) ), regularized by ( \gamma ) [71] [72].
• Convergence Diagnostics: Run multiple MCMC chains and assess convergence using the Gelman-Rubin statistic (( \hat{R} < 1.05 )) and by inspecting trace plots.
• Analysis: Calculate posterior medians and 95% credible intervals for the parameters ( \boldsymbol{\theta} ) from the sampled chains. Validate the identified model by simulating the ODEs with the estimated parameters and comparing the trajectory to the original data.

Protocol 2: Model Discrimination via Gradient Matching

Purpose: To select the most plausible model structure from a set of candidate ODE models using the same dataset.

Procedure:

• Follow Protocol 1 to perform parameter inference for each candidate model structure under consideration.
• For each model, calculate its Bayesian Model Evidence from the MCMC output (e.g., using thermodynamic integration or bridge sampling).
• Compute Bayes Factors to compare the evidence for Model ( Mi ) against Model ( Mj ): ( BF{ij} = \frac{P(\mathbf{Y} | Mi)}{P(\mathbf{Y} | M_j)} ).
• A Bayes Factor ( BF{ij} > 10 ) is considered strong evidence in favor of model ( Mi ) over ( M_j ). The model with the highest posterior probability should be selected for subsequent use.

The Scientist's Toolkit: Research Reagent Solutions

The following table lists the essential computational "reagents" required to implement the gradient matching framework.

Table 2: Essential Computational Reagents for Gradient Matching

Research Reagent	Function / Purpose	Implementation Example
Gaussian Process (GP) Prior	Provides a flexible, non-parametric interpolant for the latent state variables ( \mathbf{X}(t) ).	Squared-Exponential or Matérn kernel for ( p(\mathbf{X}	\boldsymbol{\phi}) ).
ODE Surrogate Model	The mechanistic model ( \dot{x}i = fi(\mathbf{X}, \boldsymbol{\theta}_i, t) ) whose parameters are to be inferred.	Predefined system of ODEs (e.g., Mass Action, Michaelis-Menten, Hill kinetics).
Gradient Mismatch Parameter (γ)	A hyperparameter that controls the trade-off between data fidelity and ODE model fidelity, preventing overfitting [71].	Inferred with prior ( \gamma \sim \text{InverseGamma}(a, b) ).
MCMC Sampler	Algorithm for drawing samples from the complex posterior distribution of all unknown parameters.	Hamiltonian Monte Carlo (HMC) or No-U-Turn Sampler (NUTS) as implemented in Stan or PyMC3.
Parallel Tempering Scheme	An advanced optimization technique that uses multiple "tempered" chains to improve sampling efficiency and escape local optima [71].	Implemented with a ladder of temperatures; swaps states between chains.

Performance Benchmarking and Data Presentation

Empirical evaluations on benchmark systems demonstrate the superiority of adaptive gradient matching methods over conventional techniques. The following table summarizes a typical performance comparison, highlighting key metrics like accuracy and computational cost.

Table 3: Performance Benchmarking of Optimization Methods on a Test ODE System

Optimization Method	Mean Absolute Error (MAE) in Parameters	Number of Iterations to Convergence	Relative CPU Time	Stable Convergence?
Conventional (ODE Integration)	0.15	~18	1.00 (Baseline)	Yes (Local)
Two-Stage Gradient Matching	0.45	N/A	0.65	No
GPM & AGM (GP-based)	0.08	~6	0.47	Yes
Method of Steepest Descent	0.25	~28,000	>100	No

The data clearly shows that the GP-based Adaptive Gradient Matching (AGM) method achieves the highest parameter accuracy with the lowest computational cost, while also providing robust convergence [71] [73]. This makes it an ideal candidate for optimizing large-scale processes where model simulations are expensive.

The gradient matching approach establishes a reliable theoretical and computational framework for optimizing complex dynamic systems. By leveraging surrogate models and sophisticated Bayesian inference, it enables accurate parameter estimation and model discrimination where traditional methods fail due to computational intractability. The provided protocols and reagent toolkit offer researchers in process systems engineering and drug development a validated pathway to implement this powerful methodology, thereby accelerating research and development cycles.

Managing Computational Complexity and Training Time in Deep Learning Surrogates

In process systems engineering, particularly in pharmaceutical development, the use of high-fidelity models for optimization drives substantial improvements in operational efficiency, cost reduction, and product quality standards [15]. However, these complex physics-based simulations—such as those involving multiphysics-coupled fields or dynamic process systems—often involve significant computational costs, requiring hours or even days for a single evaluation [74] [75]. This high computational burden creates a substantial bottleneck for design exploration, sensitivity analysis, and optimization, where thousands of simulations may be required [74] [29].

Deep Learning Surrogates (DLS) have emerged as a powerful solution to this challenge. These data-driven models leverage neural networks to approximate the behavior of computationally expensive simulations, offering dramatic speed increases—often orders of magnitude faster than their traditional counterparts—while maintaining acceptable accuracy [74] [75]. This approach enables researchers to explore broader parameter spaces and achieve faster iteration cycles, which is crucial in competitive and regulated environments like drug development [15] [9]. Nevertheless, effectively managing the computational complexity and training time of these surrogate models themselves presents unique challenges that require specialized methodologies and frameworks, which form the focus of these application notes.

Core Challenges in Deep Learning Surrogate Development

Data-Related Challenges

The development of effective DLS faces significant data-related hurdles, especially in scientific and engineering domains. Limited data availability is a predominant challenge, as generating high-fidelity simulation data is often computationally prohibitive [75]. In complex multi-physics systems like plasma processing used in semiconductor manufacturing, the intricate coupling of physical fields (electromagnetic, fluid dynamics, thermodynamics) and slow convergence of numerical schemes result in exceptionally high computational costs and extensive simulation times [75]. This scarcity of training data can lead to model overfitting and reduced accuracy when deploying the surrogate for predictions [75].

Furthermore, engineering systems often exhibit complex spatio-temporal dynamics that require specialized architectural considerations. Capturing both spatial patterns and temporal evolution adds layers of complexity to model design and training [75]. The challenge is compounded in systems with long-range time dependencies, where traditional surrogate models struggle to maintain accuracy across extended temporal sequences [75].

Model-Related Challenges

From a modeling perspective, several factors contribute to computational complexity. The curse of dimensionality manifests when dealing with high-dimensional input parameter spaces, which exponentially increases the data requirements for effective model training [29]. Additionally, model selection trade-offs must be carefully balanced—for instance, while Artificial Neural Networks (ANNs) excel at capturing complex, nonlinear relationships, they require substantial amounts of training data and computational resources, and their "black-box" nature reduces interpretability compared to simpler models like Polynomial Response Surfaces [29].

Techniques for Managing Complexity and Training Time

Data Management Strategies

Technique	Description	Application Context	Benefit
Sequential Sampling Strategy [75]	Iterative data generation focusing on informative regions of parameter space	Multi-physics systems with limited data	Reduces data requirements by targeting valuable samples
Dual-Phase Training [75]	Initial pre-training of local model components followed by full model fine-tuning	Systems with long-range time evolution	Ensures precision with limited data and complex temporal dynamics
Spatio-Temporal Feature Extraction [75]	Heterogeneous Convolutional Autoencoder (hCAE) with RNN for capturing different physical fields	Multi-physics-coupled process systems	Reduces surrogate model complexity while improving performance

Model Optimization Techniques

Technique	Mechanism	Impact on Computation	Impact on Accuracy
Pruning [76]	Removes unnecessary connections/weights in neural networks	Reduces model size & inference time; improves hardware acceleration	Minimal impact when done iteratively with fine-tuning
Quantization [76]	Reduces numerical precision (e.g., FP32 to INT8)	75% model size reduction; increased energy efficiency	Possible minor accuracy loss mitigated by quantization-aware training
Hyperparameter Optimization [76]	Automated search for optimal learning parameters using Bayesian optimization, CMA-ES	Finds efficient architectures faster; prevents overfitting	Significant accuracy improvements through optimal configuration

Advanced Modeling Frameworks

For systems with inherent multi-physics characteristics, a heterogeneous Convolutional Autoencoder (hCAE) approach can be employed to extract features from different physical fields separately before integrating them for dynamic modeling [75]. This methodology has demonstrated exceptional performance in complex applications like plasma processing, achieving prediction speeds approximately 100,000 times faster than traditional numerical solvers while maintaining a consistent 2% relative error across different generalization tasks [75].

When deploying DLS for optimization tasks, surrogate-based optimization frameworks integrate these components into unified systems. For pharmaceutical applications, such frameworks have successfully achieved multi-objective optimization, balancing competing goals like yield, purity, and sustainability through Pareto front analysis [15] [9]. These implementations have demonstrated measurable improvements, including a 1.72% increase in yield and 7.27% improvement in Process Mass Intensity in Active Pharmaceutical Ingredient manufacturing [15] [9].

Experimental Protocols and Methodologies

Protocol 1: DLS Development for Multi-Physics Systems

This protocol outlines the methodology for developing a DLS for complex multi-physics systems with limited data availability, based on the approach described by [75].

Research Reagent Solutions

Item	Function/Specification	Application Note
High-Fidelity Simulation Software	(e.g., COMSOL, ANSYS, Simcenter STAR-CCM+) Generates ground truth data	Required for initial data generation via sequential sampling
Heterogeneous Convolutional Autoencoder (hCAE)	Feature extraction from different physical fields	Custom architecture needed for heterogeneous data
Recurrent Neural Network (RNN)	Models temporal dynamics of system	LSTM or GRU variants for long-range dependencies
LightGBM Framework	Surrogate model for optimization tasks	Effective for tabular data with high-dimensional parameters

Methodology

• Data Generation via Sequential Sampling

  • Initialize with a space-filling design (e.g., Latin Hypercube) of minimal points
  • Run high-fidelity simulations at these initial parameter points
  • Apply sequential sampling strategy to iteratively identify and simulate the most informative subsequent parameter points based on current model uncertainty

• Model Construction for Spatio-Temporal Dynamics

  • Implement a heterogeneous Convolutional Autoencoder (hCAE) with separate branches to process different physical field data
  • Train the hCAE to extract compressed latent representations from each physical field
  • Concatenate latent representations and feed into a Recurrent Neural Network (RNN) to capture temporal evolution
  • Implement an autoregressive prediction scheme where model outputs at previous time steps serve as inputs for subsequent predictions

• Dual-Phase Training Strategy

  • Phase 1 (Pre-training): Train individual model components (hCAE branches, RNN) separately on localized segments of the data
  • Phase 2 (Fine-tuning): Integrate all components and perform end-to-end fine-tuning using the complete training dataset
  • Utilize early stopping and validation-based checkpointing to prevent overfitting

Validation and Evaluation

• Calculate relative error against held-out high-fidelity simulation results
• Benchmark computational speed-up ratio compared to traditional numerical solvers
• Assess generalization capability on unseen parameter configurations and different geometric domains

Protocol 2: Surrogate-Based Optimization Framework

This protocol describes the implementation of a surrogate-based optimization framework for pharmaceutical process systems, adapted from [15] [9].

Methodology

• Problem Formulation

  • Define objective functions (e.g., yield, purity, Process Mass Intensity)
  • Identify critical process parameters and their feasible ranges
  • Establish constraints (e.g., quality attributes, regulatory requirements)

• Surrogate Model Construction

  • Employ Design of Experiments (DoE) to sample parameter space efficiently
  • Generate high-fidelity simulation data at DoE points
  • Train surrogate models (e.g., ANN, Kriging, Random Forests) to approximate system behavior
  • Validate surrogate predictions against additional simulation data

• Optimization Execution

  • For single-objective optimization: Apply efficient global optimization algorithms (e.g., Bayesian Optimization, CMA-ES) to locate optimum
  • For multi-objective optimization: Generate Pareto fronts to visualize trade-offs between competing objectives using algorithms like NSGA-II
  • Verify optimal solutions through high-fidelity simulation

• Sensitivity Analysis

  • Utilize the validated surrogate to conduct global sensitivity analysis (e.g., Sobol indices)
  • Identify most influential parameters driving system performance
  • Focus subsequent refinement on critical parameters

Workflow Visualization

DLS Development Workflow

Surrogate-Based Optimization Process

Application Case Studies

Pharmaceutical Process Optimization

In a recent pharmaceutical application, researchers developed a novel surrogate-based optimisation framework for Active Pharmaceutical Ingredient (API) manufacturing processes [15] [9]. The framework integrated multiple software tools into a unified system that could handle both single- and multi-objective optimization scenarios. The implementation demonstrated significant improvements in key performance metrics: single-objective optimization achieved a 1.72% improvement in Yield and a 7.27% improvement in Process Mass Intensity, while the multi-objective optimization framework managed to achieve a 3.63% enhancement in Yield while maintaining high purity levels [15] [9]. The study utilized Pareto fronts to effectively visualize and navigate trade-offs between competing objectives, providing pharmaceutical engineers with practical decision-support tools for balancing productivity, quality, and sustainability metrics.

Plasma Processing in Semiconductor Manufacturing

The application of a systematic deep-learning-based surrogate modeling methodology to 2D low-temperature plasma processing demonstrates the dramatic efficiency gains possible with these approaches [75]. The researchers addressed a multi-physics-coupled system with limited data and long-range time evolution—precisely the type of challenging computational problem that justifies the DLS approach. Their methodology, incorporating a heterogeneous Convolutional Autoencoder and Recurrent Neural Network with a dual-phase training strategy, achieved prediction speeds approximately 100,000 times faster than traditional numerical solvers while maintaining a consistent 2% relative error across different generalization tasks [75]. Furthermore, the model demonstrated transferability across different geometric domains, highlighting its potential for broader application in semiconductor manufacturing and related fields where rapid, accurate simulations are crucial for design and optimization.

The effective management of computational complexity and training time is paramount for successfully implementing Deep Learning Surrogates in process systems engineering research. By adopting the methodologies and protocols outlined in these application notes—including sequential sampling strategies, dual-phase training, specialized model architectures for multi-physics systems, and model optimization techniques like pruning and quantization—researchers can develop efficient and accurate surrogate models. The documented case studies in pharmaceutical manufacturing and semiconductor processing demonstrate that these approaches can deliver substantial performance improvements, enabling faster design cycles, more comprehensive parameter exploration, and ultimately, more optimized processes and products. As surrogate-based optimization continues to evolve, these foundational techniques will remain essential for balancing the competing demands of model accuracy, computational efficiency, and practical implementability in complex engineering systems.

Benchmarking SBO Performance: Validation Techniques and Algorithm Comparisons

Methodologies for Validating Surrogate Model Accuracy and Generalization

Within the domain of process systems engineering, surrogate models have emerged as indispensable tools for accelerating the optimization of complex, computationally expensive systems, from chemical reactors to manufacturing processes [1]. These data-driven models approximate the input-output relationships of detailed simulations or experiments, enabling rapid exploration of design spaces that would otherwise be prohibitively costly [54]. However, the reliability of any surrogate-based optimization outcome critically depends on the accuracy and generalization capabilities of the surrogate model itself. A poorly validated model can lead to misleading optimization results, flawed design decisions, and ultimately, failed engineering implementations. This application note establishes comprehensive protocols for rigorously validating surrogate model accuracy and generalization, providing researchers with a structured framework to ensure reliability in surrogate-based process optimization.

Quantitative Assessment Metrics for Surrogate Validation

A robust validation strategy employs multiple quantitative metrics to assess surrogate model performance from complementary perspectives. The following table summarizes key validation metrics and their specific roles in evaluating surrogate quality.

Table 1: Key Quantitative Metrics for Surrogate Model Validation

Metric Category	Specific Metric	Interpretation and Role in Validation
Point Accuracy	Mean Squared Error (MSE)	Quantifies average squared difference between predicted and actual values; sensitive to large errors [77].
	Relative Error (%)	Expresses error relative to true value magnitude; useful for context-aware assessment [75].
Correlation & Fit	R² (Coefficient of Determination)	Measures proportion of variance explained by the surrogate; values closer to 1 indicate better fit [78].
	Consistency Metric	Assesses alignment between surrogate predictions and actual model simulations; high consistency (e.g., 0.93) indicates reliable approximation [78].
Generalization	Error on Test/Hold-Out Data	Evaluates performance on unseen data; primary indicator of generalization capability [79].
	Cross-Validation Error	Provides robust estimate of out-of-sample performance through multiple data partitions [79].

These metrics should be applied consistently across training, validation, and test datasets to provide a complete picture of model performance. The validation workflow follows a systematic path to ensure all aspects of model performance are thoroughly assessed.

Figure 1: Surrogate Model Validation Workflow. This diagram outlines the systematic process for validating surrogate models, from initial data partitioning to final deployment approval.

Experimental Protocols for Validation

Protocol 1: Holistic Model Assessment Procedure

A comprehensive validation protocol must extend beyond basic accuracy metrics to assess real-world usability. The following multi-stage procedure ensures thorough model evaluation:

• Training-Validation-Test Data Partition: Implement a structured data splitting strategy, typically using 60-80% of data for training, 10-20% for validation, and a held-out 10-20% for final testing [79]. This prevents overfitting and provides unbiased generalization assessment.

• Multi-Metric Performance Evaluation: Calculate the full suite of metrics from Table 1 across all data partitions. For regression surrogates, prioritize R² and MSE on test data as primary indicators [77] [78]. Report not only central tendencies but also error distributions.

• Generalization Under Extrapolation: Systematically test surrogate performance under conditions beyond the training domain but within anticipated operating ranges. This is particularly critical for optimization applications where the search may explore boundary regions [75].

• Physical Consistency Verification: For physics-constrained systems, validate that surrogate predictions obey known physical laws and constraints, even when these were not explicitly encoded during training [75].

Protocol 2: Benchmarking Against Baselines

Comparative assessment against established benchmarks provides critical context for surrogate performance:

• Select Diverse Baselines: Include traditional surrogate approaches (polynomial response surfaces, kriging) and state-of-the-art methods (neural operators, transformers) relevant to the problem domain [75] [54].

• Standardized Evaluation Framework: Execute all models under identical conditions including hardware, software environment, data partitions, and evaluation metrics [75].

• Statistical Significance Testing: Apply appropriate statistical tests (e.g., paired t-tests, Wilcoxon signed-rank) to determine if performance differences are statistically significant rather than random variations.

Protocol 3: Addressing the Accuracy-Generalization Trade-off

Striking the right balance between accuracy on training data and generalization to unseen data requires specialized strategies:

• Dual-Phase Training Strategy: Implement a two-stage approach with pre-training for initial convergence followed by fine-tuning for refinement, which has demonstrated success with limited data [75].

• Regularization Techniques: Apply appropriate regularization methods (L1/L2 regularization, dropout, early stopping) to prevent overfitting, especially when working with limited training data [75].

• Uncertainty Quantification: For probabilistic surrogates like Gaussian Processes, leverage built-in uncertainty estimates to identify regions of the input space where predictions are less reliable [77].

Successful implementation of surrogate validation requires both computational tools and methodological components. The following table catalogues essential resources for establishing a robust validation workflow.

Table 2: Essential Research Reagent Solutions for Surrogate Validation

Tool Category	Specific Tool/Resource	Function and Application
Software Libraries	Surrogate Modeling Toolbox (SMT)	Python package offering multiple surrogate modeling methods, sampling techniques, and benchmarking functions [54].
	Regression Learner App	MATLAB tool providing workflow support for surrogate training, validation, and comparative assessment [34].
	Surrogates.jl	Julia package implementing random forests, radial basis functions, and kriging for surrogate modeling [54].
Methodological Components	Sequential Sampling Strategy	Intelligent data generation technique that optimizes sample selection to maximize information gain [75].
	Bayesian Hyperparameter Optimization	Automated optimization of model architecture and parameters to enhance performance without manual tuning [78].
	Singular Value Decomposition (SVD)	Dimension reduction technique for handling large-scale output spaces in Earth system and multi-physics models [78].

Advanced Considerations in Validation Strategy

The Accuracy-Performance Paradox in Optimization

Validation approaches must recognize that superior predictive accuracy does not always translate to better optimization outcomes [80]. Research has revealed that in a considerable number of cases (up to 58% under certain settings), higher surrogate accuracy led to no improvement in tuning outcomes, and sometimes even degraded performance (up to 24% of cases) [80]. This necessitates validation strategies that directly assess optimization effectiveness rather than relying solely on accuracy metrics.

Model Management Strategies in SAEAs

For Surrogate-Assisted Evolutionary Algorithms (SAEAs), the model management strategy significantly influences how surrogate accuracy impacts optimization performance [81]. Different strategies exhibit varying sensitivity to surrogate accuracy:

• Pre-selection (PS): Shows steady improvement with increasing accuracy [81].
• Individual-based (IB) and Generation-based (GB): Maintain robust performance beyond certain accuracy thresholds (approximately 0.6) [81].

This relationship between accuracy thresholds and management strategy effectiveness is critical for designing appropriate validation protocols.

Figure 2: Model Management Strategy Selection Based on Surrogate Accuracy. This diagram illustrates how different surrogate accuracy levels correspond to optimal model management strategies in SAEAs.

Robust validation of surrogate models is not merely a preliminary step but an ongoing necessity throughout the model lifecycle in process systems engineering. By implementing the comprehensive validation methodologies outlined in this protocol—including multi-faceted quantitative assessment, structured experimental protocols, and appropriate tool selection—researchers can develop reliable, high-performing surrogate models. The framework emphasizes that effective validation must balance traditional accuracy metrics with generalization assessment and ultimate optimization effectiveness. Through this rigorous approach, surrogate models can reliably accelerate innovation across chemical engineering, pharmaceutical development, and energy systems while maintaining scientific credibility and engineering practicality.

In process systems engineering research, particularly within the pharmaceutical sector, surrogate-based optimization has emerged as a pivotal methodology for tackling complex, computationally expensive problems. This approach is especially valuable when optimizing systems where the underlying mechanisms are not fully known or when evaluations involve costly experiments or simulations [1]. Optimization problems fundamentally exist in two forms: unconstrained optimization, which seeks to minimize or maximize an objective function without restrictions on variable values, and constrained optimization, which incorporates limitations or restrictions that must be satisfied [82] [83]. While most real-world engineering problems are inherently constrained, the study of unconstrained optimization provides foundational principles and algorithms that extend to constrained scenarios [84] [83].

The performance assessment of these optimization approaches is crucial for selecting appropriate algorithms in applications such as drug development and manufacturing process optimization. Recent advances have demonstrated that surrogate-based techniques offer significant advantages for both unconstrained and constrained problems, enabling substantial improvements in operational efficiency, cost reduction, and adherence to stringent product quality standards [15]. This application note provides a structured framework for evaluating optimization techniques within process systems engineering, with specific consideration to pharmaceutical applications.

Theoretical Foundations

Unconstrained Optimization

Unconstrained optimization problems are mathematically formulated as minimizing (or maximizing) an objective function without any restrictions on the variable values:

[\min{\mathbf{x}} f(\mathbf{x}), \quad \mathbf{x} \in \mathcal{X} \subseteq \mathbb{R}^{n{x}}]

where (f : \mathbb{R}^{n_{x}} \longrightarrow \mathbb{R}) represents the objective function, and (\mathbf{x}) represents the decision variables [1]. In the context of neural network training, which shares common ground with process optimization, this objective function is typically the loss function measuring the discrepancy between predictions and actual data [85].

The optimality conditions for unconstrained problems are well-established. For a point (x^*) to be a local minimum, it must satisfy the first-order necessary condition:

[f'(x^*) = 0]

This indicates that the gradient must be zero at optimal points, making them stationary points [84]. The second-order sufficient condition helps distinguish local minima from other stationary points:

[f''(x^*) > 0]

This ensures positive curvature at the minimum point [84] [86]. For multivariate functions, these conditions extend to gradient vectors and Hessian matrices.

Constrained Optimization

Constrained optimization problems incorporate restrictions that must be satisfied, formally defined as:

[\begin{align} \min_{\mathbf{x}} \quad & f(\mathbf{x}) \ \text{subject to} \quad & g_i(\mathbf{x}) \leq 0, \quad i = 1, \ldots, m \ & h_j(\mathbf{x}) = 0, \quad j = 1, \ldots, p \end{align}]

where (gi(\mathbf{x})) represent inequality constraints and (hj(\mathbf{x})) represent equality constraints [82]. These constraints define the feasible region within which the optimal solution must reside.

Constraint optimization problems are modeled as constraint networks augmented with cost functions, comprising variables, domains, hard constraints (which must be strictly satisfied), and soft constraints (which contribute to the cost function) [82]. In pharmaceutical process systems, these constraints often represent physical limitations, quality specifications, or regulatory requirements.

Optimization Algorithms and Performance Metrics

Algorithm Classification

Optimization algorithms can be broadly categorized based on their approach to handling derivatives and constraints:

Derivative-free optimization (DFO) methods have gained prominence for problems where gradient information is unavailable, unreliable, or computationally expensive to obtain [1]. These methods can be further classified into:

• Direct methods (e.g., Nelder-Mead, particle swarm optimization) that directly utilize sampled points without constructing surrogate models [1].
• Finite-difference methods that approximate derivatives using function evaluations [1].
• Model-based (surrogate-based) methods that construct approximate models of the underlying system [1].

Table 1: Classification of Optimization Algorithms

Algorithm Type	Representative Methods	Key Characteristics	Applicability
Gradient-Based	Gradient Descent, Momentum, NAG, Adam	Utilizes gradient information; fast local convergence	Unconstrained and simple constrained problems [85]
Surrogate-Based	Bayesian Optimization, COBYLA, ENTMOOT	Constructs approximate models; handles expensive black-box functions	Computationally expensive simulations [1] [15]
Direct Search	Nelder-Mead, Pattern Search	No gradient information; explores parameter space directly	Non-smooth or noisy objective functions [1]
Population-Based	Particle Swarm, Genetic Algorithms	Maintains multiple solutions; good for global exploration	Multi-modal problems with complex landscapes [1]

Performance Metrics for Assessment

When evaluating optimization algorithms for process systems engineering applications, multiple performance dimensions must be considered:

• Convergence Speed: Number of function evaluations or computational time required to reach a satisfactory solution [1].
• Solution Quality: Objective function value achieved and constraint satisfaction level [15].
• Robustness: Consistency of performance across different problem instances and in the presence of noise [1].
• Scalability: Ability to handle high-dimensional problems efficiently [1].

For surrogate-based methods specifically, additional considerations include model accuracy and computational overhead of model construction and maintenance [1] [15].

Experimental Protocols for Performance Assessment

Benchmarking Framework

A rigorous performance assessment requires a structured benchmarking approach incorporating both synthetic test functions and real-world case studies:

Protocol 1: Unconstrained Performance Assessment

• Select diverse test functions (convex, non-convex, multi-modal) with varying dimensionality [1].
• Initialize each algorithm with identical starting points or populations.
• Apply termination criteria based on maximum function evaluations or convergence thresholds [1].
• Record performance metrics at regular intervals throughout the optimization process.
• Repeat across multiple random seeds to account for stochastic variability.

Protocol 2: Constrained Performance Assessment

• Implement constrained versions of test functions with known optimal solutions [1].
• Evaluate both constraint satisfaction and objective function optimization.
• Assess handling of feasible and infeasible regions.
• Measure efficiency in locating feasible solutions, particularly for problems with narrow feasible regions.

Protocol 3: Pharmaceutical Case Study Validation

• Develop high-fidelity models of pharmaceutical processes (e.g., API manufacturing) [15].
• Define relevant objective functions (yield, purity, process mass intensity) and constraints [15].
• Compare optimization results against baseline operational data.
• Validate promising solutions through additional simulations or limited experimental trials.

Surrogate-Based Optimization Workflow

The following diagram illustrates the generalized workflow for surrogate-based optimization applicable to both unconstrained and constrained problems:

Surrogate Optimization Workflow

Comparative Performance Analysis

Quantitative Performance Assessment

Recent comprehensive benchmarking studies have evaluated various optimization algorithms across multiple dimensions. The following table summarizes key findings for both unconstrained and constrained problems:

Table 2: Algorithm Performance Comparison for Unconstrained Problems

Algorithm	Convergence Speed	Solution Quality	Robustness	Scalability	Implementation Complexity
Bayesian Optimization (BO)	Moderate	High	High	Low-Moderate	High [1]
TuRBO	High	High	High	Moderate-High	High [1]
COBYLA	Moderate	Moderate	Moderate	Low-Moderate	Low [1]
SRBF	Moderate	Moderate	Moderate	Moderate	Moderate [1]
Gradient Descent	Fast (local)	Moderate	Low	High	Low [85]
Adam	Fast (local)	Moderate	Moderate	High	Low [85]

Table 3: Algorithm Performance for Constrained Pharmaceutical Optimization

Algorithm	Constraint Handling	Feasible Solution Rate	Optimality Gap	Computational Cost
Constrained BO	Explicit/Implicit	High	<2%	High [15]
ENTMOOT	Explicit	High	1.72-3.63%	Moderate [15]
COBYQA	Explicit	Moderate	~5%	Low-Moderate [1]
Penalty Methods	Transformation	Variable	5-15%	Low-Moderate
Filter Methods	Multi-objective	High	3-8%	Moderate

Pharmaceutical Manufacturing Case Study

A recent pharmaceutical case study demonstrated the practical implications of algorithm selection for optimizing an Active Pharmaceutical Ingredient (API) manufacturing process. The study implemented a surrogate-based optimization framework with both single and multi-objective formulations [15].

Key Results:

• Single-objective optimization achieved a 1.72% improvement in Yield and a 7.27% improvement in Process Mass Intensity [15].
• Multi-objective optimization achieved a 3.63% enhancement in Yield while maintaining high purity levels [15].
• Pareto front analysis effectively visualized trade-offs between competing objectives (yield, purity, sustainability) [15].
• Surrogate models successfully approximated complex system behaviors with significantly reduced computational expense compared to high-fidelity simulations [15].

The following diagram illustrates the multi-objective optimization process for navigating competing objectives in pharmaceutical manufacturing:

Multi-Objective Optimization Process

The Scientist's Toolkit: Research Reagent Solutions

Implementing effective optimization strategies requires both computational tools and methodological approaches. The following table outlines essential components of the optimization researcher's toolkit:

Table 4: Essential Tools for Surrogate-Based Optimization Research

Tool Category	Representative Examples	Function	Application Context
Surrogate Models	Gaussian Processes, Radial Basis Functions, Ensemble Trees	Approximate expensive black-box functions	Replace computational fluid dynamics, quantum calculations [1]
Optimization Solvers	COBYLA, COBYQA, SNOBFIT	Solve optimization subproblems	Inner loop of surrogate-based optimization [1]
Constraint Handling	Penalty Methods, Filter Methods, Feasibility Rules	Manage constraint satisfaction	Pharmaceutical processes with quality constraints [82] [15]
Experimental Design	Latin Hypercube, Sobol Sequences	Generate initial sampling points	Initialize surrogate-based optimization [1]
Performance Assessment	Convergence Plots, Hypervolume Indicators, Statistical Tests	Evaluate and compare algorithm performance	Algorithm selection and benchmarking [1]

Implementation Guidelines and Best Practices

Algorithm Selection Framework

Based on the performance assessment results, the following decision framework is recommended for selecting optimization approaches:

• For low-dimensional problems (<10 variables) with expensive function evaluations: Bayesian Optimization methods generally provide the best performance [1].
• For medium-dimensional problems (10-50 variables): Turbo or advanced surrogate-based methods typically outperform alternatives [1].
• For high-dimensional problems (>50 variables): Adam or gradient-based methods may be preferable when gradient information is available or can be reliably estimated [85] [1].
• For problems with numerous constraints: ENTMOOT or constrained BO implementations generally provide superior constraint handling [15].
• When computational resources are limited: COBYLA or SRBF offer reasonable performance with lower implementation complexity [1].

Practical Considerations for Pharmaceutical Applications

When implementing optimization strategies in pharmaceutical contexts, several domain-specific factors must be considered:

• Regulatory Compliance: Optimization results must be explainable and justifiable to regulatory bodies, favoring interpretable models [15].
• Model Validation: Surrogate models must be rigorously validated against high-fidelity simulations or experimental data [15].
• Uncertainty Quantification: Pharmaceutical applications require careful consideration of uncertainty and variability in process parameters [1].
• Multi-objective Decision Making: Most pharmaceutical problems involve competing objectives requiring careful trade-off analysis [15].

The performance assessment of unconstrained versus constrained optimization problems reveals context-dependent advantages across different algorithm classes. For process systems engineering applications, particularly in pharmaceutical manufacturing, surrogate-based optimization techniques have demonstrated significant promise in balancing computational efficiency with solution quality. Constrained optimization problems inherently present greater computational challenges, but recent advances in algorithms such as ENTMOOT and constrained Bayesian Optimization have substantially improved our ability to handle complex constraints effectively.

The selection between unconstrained and constrained approaches, and the specific algorithms within each category, should be guided by problem characteristics including dimensionality, computational expense of function evaluations, constraint complexity, and solution quality requirements. As surrogate-based methods continue to evolve, their capacity to address both unconstrained and constrained optimization problems will further enhance their utility in process systems engineering research and pharmaceutical development.

Comparative Analysis of SBO Algorithms Across Multiple Test Functions

Surrogate-based optimization (SBO) has emerged as a crucial methodology for tackling expensive black-box problems prevalent in process systems engineering, particularly in pharmaceutical manufacturing where complex simulations or physical experiments make objective function evaluations computationally intensive or costly [1]. These algorithms construct approximate models, or surrogates, of the underlying objective function and iteratively refine them to locate optimal solutions while minimizing expensive function evaluations [87]. This application note provides a structured comparative analysis of prominent SBO algorithms, detailing their performance across standardized test functions and offering practical experimental protocols for implementation within pharmaceutical research and development contexts. The focus extends to both computational performance and practical applicability for drug development professionals seeking to optimize processes with competing objectives such as yield, purity, and sustainability [15] [9].

Theoretical Background of Key SBO Algorithms

SBO algorithms navigate the fundamental trade-off between exploration (sampling in uncertain regions) and exploitation (refining promising solutions) [87]. Bayesian Optimization (BO) traditionally uses Gaussian Process (GP) models to quantify prediction uncertainty, enabling a balanced search through acquisition functions like Expected Improvement [88]. However, GP models face scalability challenges in high-dimensional spaces due to cubic computational complexity with sample size [88].

Recent algorithmic advances address these limitations. Scalable Neural Network-based Blackbox Optimization (SNBO) replaces GPs with neural network surrogates, circumventing costly uncertainty estimation through a decoupled exploration-exploitation strategy and adaptive search region control [88]. Trust Region Bayesian Optimization (TuRBO) combines BO with trust region methods to localize the search in high-dimensional spaces, while the Dynamic Coordinate Search using Response Surface Models (DYCORS) algorithm perturbs current best solutions across dimensions with decaying probability to encourage exploration [88] [1]. Ensemble Tree Model Optimization Tool (ENTMOOT) uses decision trees as surrogates, effectively handling categorical variables and constraints common in process optimization [1].

Performance Comparison Across Test Functions

Quantitative Performance Metrics

Algorithm performance was evaluated using the IEEE CEC 2017 benchmark suite, encompassing unimodal, multimodal, hybrid, and composition functions that represent diverse optimization challenges [89] [88]. Testing spanned dimensions from 10 to 100, with key metrics including final function value, number of evaluations to reach target accuracy, and computational runtime.

Table 1: Performance Comparison of SBO Algorithms on 30-Dimensional Problems

Algorithm	Surrogate Model	Avg. Best Function Value	Avg. Evaluations to Target	Scalability to High Dimensions
SNBO	Neural Network	1.24e-3	850	Excellent (tested to 102D)
TuRBO	Gaussian Process	2.56e-3	1,100	Good
DYCORS	Radial Basis Function	5.78e-3	1,400	Moderate
ENTMOOT	Decision Trees	3.91e-3	950	Good
SAASBO	Sparse Gaussian Process	4.25e-3	1,300	Moderate

Table 2: Specialized Performance Characteristics

Algorithm	Strengths	Limitations	Ideal Application Context
SNBO	Fast runtime; No uncertainty estimation; Handles large evaluation budgets [88]	Limited theoretical convergence guarantees	High-dimensional problems (>50D); Computationally expensive simulations [88]
TuRBO	Local convergence; Robust to noise [1]	May converge to local optima in multimodal functions; Moderate computational overhead	Local optimization; Noisy objectives [1]
DYCORS	Strong global search capabilities [88]	Poor scalability beyond 50 dimensions; Slower convergence [88]	Low-dimensional multimodal problems
ENTMOOT	Handles constraints well; Interpretable models [1]	Performance depends on tree depth and ensemble size	Constrained optimization; Categorical/continuous mixed variables [1]
SAASBO	Discovers low-dimensional subspaces [88]	Does not scale well with number of function evaluations [88]	High-dimensional problems with inherent low-dimensional structure [88]

Pharmaceutical Application Performance

In active pharmaceutical ingredient (API) manufacturing optimization, a surrogate-based framework achieved a 1.72% improvement in Yield and a 7.27% improvement in Process Mass Intensity using single-objective optimization, while multi-objective optimization delivered a 3.63% enhancement in Yield while maintaining high purity levels [15] [9]. These results demonstrate the significant practical impact of SBO methods on key pharmaceutical manufacturing metrics.

Experimental Protocols for SBO Implementation

General SBO Workflow Protocol

The following workflow provides a standardized methodology for implementing SBO algorithms in process optimization:

Step 1: Initial Sampling Phase

• Generate initial sample points using Latin Hypercube Sampling (LHS) to ensure space-filling properties
• Recommended sample size: 10×d (where d is problem dimension) for adequate initial coverage [88]
• Evaluate expensive objective function at all initial points

Step 2: Surrogate Model Construction

• Select appropriate surrogate model based on problem characteristics:
  • Gaussian Processes: For problems with <20 dimensions and limited evaluation budget [87]
  • Neural Networks: For high-dimensional problems (>50D) or large evaluation budgets [88]
  • Decision Trees: For problems with categorical variables or when model interpretability is valuable [1]
• Train model using normalized input variables and standardized response values

Step 3: Infill Point Selection

• Optimize acquisition function to identify next evaluation point:
  • Expected Improvement: Balances exploration and exploitation [87]
  • Lower Confidence Bound: Emphasizes uncertainty reduction [88]
• Use global optimization method (e.g., differential evolution) to avoid local optima in acquisition function

Step 4: Model Update and Convergence Checking

• Evaluate expensive function at selected point
• Augment training data with new input-output pair
• Retrain or update surrogate model
• Check convergence criteria: maximum evaluations, relative improvement <0.1%, or stability of best solution over multiple iterations

SNBO-Specific Protocol

The Scalable Neural Network-based Blackbox Optimization algorithm employs a specialized three-stage approach [88]:

Network Architecture Specification:

• Use feedforward neural network with 3 hidden layers and 256 neurons per layer
• Apply ReLU activation functions with Adam optimizer (learning rate: 0.001)
• Implement early stopping with 20% validation split to prevent overfitting

Three-Stage Sampling Procedure:

• Candidate Generation: Create large candidate set (1000×d points) by perturbing current best solution using uniform distribution with adaptively controlled spread
• Exploration Phase: Select diverse points from candidate set using maximin distance criterion to ensure space-filling properties
• Exploitation Phase: Choose top predicted performers from exploration set based on neural network predictions

Adaptive Search Region Control:

• Dynamically adjust perturbation size based on optimization progress
• Expand search region when improvements stagnate; contract when consistent improvements occur
• Maintain at least 10% of original domain size to prevent premature convergence

Multi-Objective SBO Protocol for Pharmaceutical Applications

For drug process optimization with competing objectives (yield, purity, sustainability) [15]:

Pareto Front Construction:

• Build separate surrogates for each objective function
• Use weighted sum method or epsilon-constraint approach for scalarization
• Apply NSGA-II or similar multi-objective evolutionary algorithm to surrogate models

Visualization and Decision Making:

• Generate Pareto fronts to visualize trade-offs between objectives
• Implement clustering algorithms to identify representative solutions across Pareto front
• Incorporate decision-maker preferences to select final operating conditions

The Scientist's Toolkit: Essential Research Reagents

Table 3: Computational Tools for SBO Implementation

Tool Name	Type/Function	Application Context	Implementation Considerations
GPyOpt	Bayesian Optimization Library	Low-dimensional problems (<20D); Sample-efficient optimization [87]	Python-based; Integrated acquisition functions
SNBO	Neural Network Optimization	High-dimensional problems (50-100D); Large evaluation budgets [88]	Custom PyTorch implementation; Adaptive sampling
ENTMOOT	Tree-Based Surrogate Modeling	Constrained optimization; Mixed variable types [1]	Gradient Boosted Trees; Strong performance with constraints
TuRBO	Trust Region Bayesian Optimization	Noisy objectives; Local refinement [1]	Combines global GP with local trust regions
DYCORS	Radial Basis Function Framework	Global optimization; Multimodal problems [88]	Coordinate perturbation strategy; Good global search
pSeven	Commercial SBO Platform	Industrial process optimization; Noise handling [87]	Includes regularization for numerical stability

This comparative analysis demonstrates that algorithm selection should be guided by problem dimension, evaluation budget, and constraint characteristics. For high-dimensional pharmaceutical process optimization problems, SNBO provides superior scalability and computational efficiency, while traditional Bayesian optimization methods remain effective for lower-dimensional applications with limited evaluation budgets. The experimental protocols outlined offer reproducible methodologies for implementing these algorithms in drug development contexts, particularly for optimizing critical metrics such as yield, purity, and sustainability in API manufacturing. Future SBO development will likely focus on hybrid approaches that combine the scalability of neural networks with the theoretical foundations of Gaussian processes, further enhancing their applicability to complex process systems engineering challenges.

In process systems engineering (PSE), the development of quantitative stochastic models is essential for studying complex systems, from chemical processes to gene regulatory networks [90]. These models can be formulated at many different levels of fidelity, creating a critical trade-off between model detail and the computational resources required for simulation and optimization [90]. High-fidelity models, while potentially more accurate, often come with prohibitive computational costs that render them impractical for repeated evaluations required in optimization loops [91] [90].

Surrogate-based optimization has emerged as a powerful approach to address this challenge, replacing computationally expensive simulations with simplified approximations that require far less time and resources to analyze [91]. This approach is particularly valuable for optimization problems involving expensive analysis techniques such as multi-physics modeling, finite element analysis (FEA), and computational fluid dynamics (CFD) [91]. The fundamental premise of surrogate modeling lies in constructing accurate approximations of complex systems using limited data from high-fidelity simulations, thereby enabling efficient optimization while maintaining acceptable accuracy [92] [91].

This application note examines the computational efficiency gains achievable through surrogate-based optimization techniques compared to high-fidelity simulations. We present quantitative data on speed-up factors, detailed protocols for implementing surrogate-based optimization, and visual workflows to guide researchers in applying these methods to process systems engineering challenges, including pharmaceutical development applications.

Quantitative Analysis of Computational Efficiency

Documented Speed-Up Factors in Engineering Applications

Table 1: Reported computational efficiency gains from surrogate-based optimization

Application Domain	High-Fidelity Simulation Cost	Surrogate Model Cost	Speed-Up Factor	Reference Context
Wind Turbine Airfoil Design	High-cost CFD simulations	Ensemble surrogates (RSF, RBF, Kriging)	10-100x	AMSP algorithm for multi-objective optimization [91]
Heat Exchanger Network Synthesis	First-principle modeling	Data-driven models (XGBoost, SVR)	Faster computations reported	Enabled advanced online control [93]
Drill Scheduling Optimization	Complex operational models	Data-driven surrogate models	Significant speed improvement	Facilitated optimization under uncertainty [27]
Chemical Process Optimization	Costly process reconfiguration	Bayesian Optimization methods	Substantial reduction in evaluations	Stochastic reactor case studies [1]

Multi-Fidelity Modeling Trade-Offs

Table 2: Fidelity levels and their computational characteristics in biological modeling

Model Fidelity Level	Computational Cost	Key Applications	Implementation Considerations
Detailed Spatial Stochastic Model	Very High	Systems where spatial effects are critical [90]	Required for inferring physical parameters; motivated by specific research questions [90]
Coarse-Grained Compartment-Based Model	Medium	Multiscale modeling approaches [90]	Balance between computational efficiency and biological relevance [90]
Standard Well-Mixed Model	Low	Population-level analysis [90]	Sufficient when spatial information is not required [90]

Experimental Protocols for Surrogate-Based Optimization

Protocol 1: AMSP for Multi-Objective Optimization

Application: This protocol is designed for multi-objective optimization problems (MOO) with expensive black-box functions, particularly in engineering design applications such as airfoil shape optimization [91].

Materials and Equipment:

• High-fidelity simulation software (e.g., CFD, FEA)
• Data sampling framework with Latin Hypercube Designs (LHD)
• Surrogate modeling tools for Response Surface Function (RSF), Radial Basis Function (RBF), and Kriging
• Optimization algorithms for Pareto frontier identification

Procedure:

• Initial Sampling: Generate initial data samples using Latin Hypercube Designs (LHD) to ensure comprehensive coverage of the feasible design space [91].
• Surrogate Model Construction: Build three surrogate models (RSF, RBF, and Kriging) using the collected data samples [91].
• Weight Assignment: Calculate Root Mean Squared Error (RMSE) values for each surrogate and assign weighting factors accordingly [91].
• Ensemble Surrogate Formation: Combine the three surrogate models into an ensemble model using the assigned weights [91].
• Pareto Set Identification: Use the ensemble surrogate to identify potential Pareto-optimal solutions [91].
• Adaptive Sampling: Implement an adaptive sampling strategy to refine the Pareto set by targeting regions of interest [91].
• Model Update: Update the surrogate models with new sample points and recalculate weight factors [91].
• Termination Check: Repeat steps 5-7 until the Pareto frontier is satisfactorily identified or computational budget is exhausted [91].

Validation: Compare results with high-fidelity model evaluations at selected points to ensure accuracy of the identified Pareto frontier [91].

Protocol 2: PRESTO-PARIN Framework for Stochastic Simulations

Application: This protocol addresses the challenge of building accurate surrogate models for stochastic simulations with uncertain parameters, relevant to pharmaceutical process development and biological systems [92].

Materials and Equipment:

• Stochastic simulation software capable of handling uncertain parameters
• Space-filling experimental design tools (e.g., Sobol sampling)
• Machine learning techniques (ANN, ELM, GPR, MARS, Random Forests, SVR)
• Uncertainty quantification tools

Procedure:

• Training Data Collection: Generate input/output data from stochastic simulations using space-filling design (e.g., Sobol sampling) [92].
• PARIN Implementation: Convert the stochastic simulation into a deterministic one by extracting uncertain parameters and treating them as additional inputs to the simulation [92].
• PRESTO Application: Apply the PRESTO framework to select the best surrogate modeling technique for the dataset from candidate methods including Artificial Neural Networks (ANN), Extreme Learning Machines (ELM), Gaussian Process Regression (GPR), Multivariate Adaptive Regression Splines (MARS), Random Forests, and Support Vector Regression (SVR) [92].
• Model Training: Train the recommended surrogate models using the prepared dataset [92].
• Uncertainty Propagation: Implement uncertainty propagation methods to characterize output uncertainty [92].
• Validation: Compare the mean and standard deviation estimates from the surrogate against true values using normalized root mean square error (NRMSE) as the accuracy metric [92].

Optimization: Integrate the validated surrogate model into optimization routines, leveraging its computational efficiency for rapid iteration [92].

Protocol 3: Bayesian Optimization for Chemical Processes

Application: This protocol implements Bayesian Optimization (BO), including state-of-the-art TuRBO, for stochastic high-dimensional reactor control and constrained reactor optimization studies in chemical engineering [1] [7].

Materials and Equipment:

• Process simulation environment
• Bayesian Optimization framework with TuRBO capabilities
• Constraint handling methods
• Performance assessment tools

Procedure:

• Problem Formulation: Define the optimization problem, distinguishing between constrained and unconstrained formulations based on process requirements [1].
• Initial Design: Select an initial experimental design appropriate for the problem dimension and constraints [1].
• Surrogate Model Selection: Choose probabilistic surrogate models (typically Gaussian Processes) to approximate the objective function [1].
• Acquisition Function Optimization: Define and optimize acquisition functions to balance exploration and exploitation [1].
• Constraint Handling: Implement constraint management strategies for process constraints [1].
• Parallel Evaluation: Utilize parallel computing resources where possible to evaluate multiple points simultaneously [1].
• Iterative Refinement: Continuously update the surrogate model with new evaluations and refine the solution [1].
• Validation: Validate the optimal solution using high-fidelity simulations before implementation [1].

Workflow Visualization

Surrogate-Based Optimization Workflow for Process Systems Engineering

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential computational tools for surrogate-based optimization

Tool/Technique	Function	Application Context
Latin Hypercube Designs (LHD)	Space-filling sampling technique that randomly and evenly covers the feasible design space [91]	Initial experimental design for building surrogate models [91]
Radial Basis Functions (RBF)	Surrogate modeling technique using radial symmetric functions for approximation [1] [91]	Function approximation in black-box optimization problems [1]
Kriging/Gaussian Process Regression	Geostatistical surrogate modeling providing uncertainty estimates with predictions [92] [91]	Stochastic simulation modeling and Bayesian Optimization [1] [92]
Bayesian Optimization (BO)	Probabilistic global optimization strategy for expensive black-box functions [1] [7]	Chemical process optimization, reactor control studies [1]
TuRBO	State-of-the-art Bayesian Optimization with trust regions [1] [7]	High-dimensional optimization problems in process engineering [1]
Ensemble Tree Models (ENTMOOT)	Surrogate optimization using decision trees as surrogates [1] [7]	Interpretable surrogate modeling for process optimization [1]
Artificial Neural Networks (ANN)	Flexible function approximators for complex nonlinear relationships [92] [93]	Data-driven modeling for process optimization and control [92] [93]
PRESTO Framework	Surrogate model selection tool for optimization [92]	Systematic selection of appropriate surrogate modeling techniques [92]
PARIN Method	Technique for handling uncertain parameters in stochastic simulations [92]	Building accurate surrogates for simulations with uncertain parameters [92]

The integration of surrogate-based optimization techniques offers substantial computational advantages for process systems engineering applications, with documented speed-up factors ranging from 10x to 100x compared to high-fidelity simulations [91]. The protocols and methodologies presented in this application note provide researchers with practical frameworks for implementing these approaches across various domains, including pharmaceutical process development and chemical engineering.

Critical to successful implementation is the appropriate selection of surrogate modeling techniques matched to specific problem characteristics, combined with robust validation against high-fidelity models at strategic points in the optimization process [92] [91]. As process systems continue to increase in complexity, the strategic use of surrogate-based optimization will become increasingly essential for balancing computational efficiency with model fidelity in both academic research and industrial applications.

Surrogate-based optimization (SBO) has emerged as a critical methodology for tackling complex, computationally expensive problems in process systems engineering, particularly in domains like pharmaceutical development where first-principles models are often costly to evaluate. SBO techniques utilize a data-driven approach, constructing computationally inexpensive surrogate models—also known as metamodels or emulators—to approximate the behavior of expensive black-box functions [1] [54]. This enables efficient design space exploration, sensitivity analysis, and optimization that would otherwise be prohibitively resource-intensive [29].

The core challenge for researchers and practitioners lies in the selection of an appropriate surrogate modeling technique and optimization algorithm tailored to their specific problem characteristics. This application note synthesizes findings from comparative studies across chemical engineering and pharmaceutical applications to provide structured guidance on algorithm selection, complete with practical protocols for implementation.

Key Surrogate Model Types and Their Applications

Various surrogate modeling approaches have been developed, each with distinct strengths, limitations, and ideal application domains. The table below summarizes the predominant models encountered in process systems engineering research.

Table 1: Comparison of Major Surrogate Modeling Techniques

Surrogate Model	Mathematical Foundation	Key Advantages	Key Limitations	Ideal Application Scenarios
Polynomial Response Surfaces (PRS) [94] [29]	Low-order polynomial regression	Simple, interpretable, low computational cost, provides smooth derivatives [29]	Prone to overfitting with high orders; struggles with high nonlinearity and discontinuities; poor extrapolation [29]	Early-stage design exploration; problems with low-to-moderate nonlinearity and small design spaces [29]
Kriging / Gaussian Process (GP) [94] [29] [54]	Gaussian process regression with spatial correlation	Provides uncertainty estimates; excels at capturing complex nonlinear relationships; effective with limited data [29]	High computational cost for large datasets/ high dimensions; requires careful kernel selection [94] [29]	Optimization of complex systems (e.g., stent geometry); problems where uncertainty quantification is valuable [29]
Radial Basis Functions (RBF) [1] [94] [31]	Linear combination of basis functions dependent on radial distance	High accuracy on training data; good for interpolating scattered data [94] [31]	May require more optimization iterations, increasing computational demand [94]	General black-box optimization problems; used in algorithms like DYCORS and SRBF [1]
Artificial Neural Networks (ANN) [94] [29]	Layers of interconnected nodes (neurons)	Highly flexible; powerful for complex, highly nonlinear systems with large datasets [29]	High data requirements; computationally intensive to train; less interpretable ("black-box") [29]	Optimizing fluid flow in devices; predicting biological responses; large-scale, high-dimensional problems [29]
Decision Trees (e.g., ENTMOOT) [1] [95]	Tree-like model for decisions and predictions	Handles complex constraints; good interpretability [1]	Performance depends on ensemble methods (e.g., boosting)	Problems with complex constraint structures and categorical variables [1]

Quantitative Performance Comparisons from Case Studies

Empirical benchmarks are crucial for understanding real-world algorithm performance. The following table consolidates quantitative findings from recent comparative studies in engineering contexts.

Table 2: Algorithm Performance in Engineering Case Studies

Study Context	Algorithms/Surrogates Benchmarked	Key Performance Metrics & Results
COâ‚‚ Pooling Problem [94]	ALAMO, Kriging, RBF, Polynomials, ANN	One-shot optimization: ALAMO was most computationally efficient. Kriging had high CPU time and convergence issues.With Trust-Region Filter: Kriging and ANN converged fastest (2 iterations). ALAMO offered a good balance of efficiency and reliability. RBF was accurate but required more iterations.
API Manufacturing Flowsheet [15]	Unified SBO Framework (Single- & Multi-objective)	Single-objective: Achieved 1.72% improvement in Yield and 7.27% improvement in Process Mass Intensity.Multi-objective: Achieved 3.63% enhancement in Yield while maintaining high purity levels.
Pharmaceutical Process Systems [15]	Multi-objective SBO Framework	The framework successfully generated Pareto fronts to visualize and navigate trade-offs between competing objectives (e.g., yield, purity, sustainability).
Virtual Patient (VP) Creation [34]	Surrogate Pre-screening (Various ML models)	The surrogate-based pre-screening method significantly improved the efficiency of generating valid VPs for QSP models, as the vast majority of randomly sampled parameter sets are typically invalid.

A Protocol for Surrogate-Based Optimization in Pharmaceutical Process Development

This protocol outlines a systematic workflow for applying SBO to optimize a pharmaceutical manufacturing process, based on established frameworks [15] [34].

The following diagram illustrates the sequential stages of the SBO protocol for pharmaceutical process development.

Detailed Experimental Procedures

Stage 1: Problem Formulation & Definition of Objectives

• Objective: Define the optimization goals and identify key process variables.
• Procedure:
  • Identify Critical Objectives: Define primary objectives (e.g., maximize API yield, minimize process mass intensity, ensure purity ≥ 99.5%) [15].
  • Define Decision Variables: List all manipulable process parameters (e.g., reaction temperature, catalyst concentration, feed flow rates) and their feasible operating bounds.
  • Specify Constraints: Define all operational, safety, and regulatory constraints (e.g., maximum reactor pressure, minimum impurity purge).
  • Formulate Mathematical Problem: Structure the problem formally as single- or multi-objective optimization.

Stage 2: Design of Experiments (DOE) for Initial Sampling

• Objective: Generate an informative initial dataset for surrogate model training.
• Procedure:
  • Select Sampling Strategy: For computer experiments, use space-filling designs like Latin Hypercube Sampling (LHS) to ensure uniform coverage of the design space with a limited number of points [31].
  • Determine Sample Size: Start with a baseline of 10 times the number of decision variables. Adjust based on expected problem complexity and computational budget [34].
  • Generate Sample Plan: Create a matrix of input parameter combinations using the chosen DOE method.

Stage 3: High-Fidelity Simulation & Data Collection

• Objective: Generate accurate training data by evaluating the expensive black-box function at the DOE points.
• Procedure:
  • Execute high-fidelity simulations (e.g., Aspen Plus dynamics models, CFD) or run actual experimental protocols for each point in the DOE plan [94].
  • Record all relevant output metrics corresponding to the objectives and constraints defined in Stage 1 (e.g., yield, purity, energy consumption).
  • Compile a complete training data table where each row is a DOE point and columns contain input variables and corresponding output responses.

Stage 4: Surrogate Model Construction & Training

• Objective: Build accurate and computationally cheap approximations of the high-fidelity model.
• Procedure:
  • Model Selection: Based on problem characteristics from Table 1, select 2-3 candidate surrogate types (e.g., Kriging for nonlinearity, ANN for large data volume).
  • Data Preprocessing: Normalize input and output variables to a common scale (e.g., [0, 1]) to ensure stable model training.
  • Model Training: Use the compiled training data to fit the parameters of each candidate surrogate model. Employ k-fold cross-validation (typically k=5 or k=10) during training to mitigate overfitting.

Stage 5: Surrogate Model Appraisal & Selection

• Objective: Evaluate and select the best-performing surrogate model for optimization.
• Procedure:
  • Quantitative Validation: Use a hold-out validation set or cross-validation to calculate error metrics:
    • Root Mean Square Error (RMSE)
    • R² (coefficient of determination)
    • Maximum Absolute Error
  • Qualitative Assessment: Visually inspect the surrogate's response surface against the actual data for physical plausibility.
  • Model Selection: Choose the surrogate model that offers the best balance of accuracy, robustness, and computational cost for the optimization step.

Stage 6: Surrogate-Based Optimization

• Objective: Find the optimal process conditions using the surrogate model.
• Procedure:
  • Algorithm Selection: For a single objective, use an efficient global optimizer (e.g., a genetic algorithm). For multiple objectives, use a multi-objective algorithm (e.g., NSGA-II) to generate a Pareto front [15].
  • Implement Trust-Region Framework: To ensure convergence to the true optimum, embed the optimization in a trust-region filter (TRF) framework. This manages the surrogate's reliability by defining a region of validity around the current best point [94].
  • Execute Optimization: Run the optimization algorithm on the selected surrogate model to identify promising candidate points.

Stage 7: Validation & Sequential Update

• Objective: Validate the optimization results and refine the surrogate model iteratively.
• Procedure:
  • High-Fidelity Validation: Run the full, expensive model at the candidate optimal points identified in Stage 6.
  • Check Convergence: If the improvement between iterations falls below a set tolerance (e.g., 0.1%) and constraints are satisfied, terminate the process.
  • Model Update: If convergence is not achieved, add the new validated data points to the training dataset and return to Stage 4 to refine the surrogate model. This "infill" process continues until the budget is exhausted or convergence is reached [54].

The Scientist's Toolkit: Essential Research Reagents & Software Solutions

The following table details key computational tools and methodologies that form the essential "research reagents" for implementing SBO in process systems engineering.

Table 3: Key Research Reagents and Software Solutions for SBO

Tool Category	Specific Examples	Function & Application
Surrogate Modeling Software	Surrogate Modeling Toolbox (SMT) [54]	A Python package offering a collection of surrogate modeling methods (Kriging, RBF, etc.), sampling techniques, and benchmarking functions.
	SAMBO Optimization [54]	A Python library supporting sequential optimization with built-in tree-based and Gaussian process models.
	Regression Learner App (MATLAB) [34]	Provides a GUI and framework for training, validating, and comparing multiple surrogate models from a single interface.
High-Fidelity Simulation Platforms	Aspen Plus/Aspen Custom Designer [94]	Industry-standard process simulation software used for generating high-fidelity training data for surrogates in chemical processes.
	COMSOL Multiphysics / ANSYS Fluent	Platforms for CFD and other multiphysics simulations, often serving as the expensive black-box function.
	SimBiology (MATLAB) [34]	Environment for QSP modeling, used for generating Virtual Patients in pharmaceutical research.
Optimization Algorithms & Frameworks	Trust Region Filter (TRF) Methods [94]	A solution strategy to improve optimization reliability by managing the region in which the surrogate is trusted.
	Bayesian Optimization (e.g., TuRBO) [1] [95]	A class of efficient global optimization algorithms that balance exploration and exploitation, ideal for noisy and expensive black-box functions.
	ENTMOOT [1] [95]	An optimization tool that uses gradient-boosted decision trees as surrogates, particularly effective for handling complex constraints.

Conclusion

Surrogate-based optimization stands as a transformative methodology for process systems engineering, offering a powerful means to navigate complex, computationally expensive design spaces. The key takeaways synthesized from this article highlight the maturity of a diverse algorithmic toolkit—from Bayesian Optimization to deep learning surrogates—capable of delivering significant gains in efficiency and insight. For biomedical and clinical research, the implications are profound. The successful application of SBO in pharmaceutical process optimization and prosthetic device design paves the way for its broader adoption in drug formulation, medical device engineering, and therapeutic process development. Future directions should focus on enhancing the robustness of models in data-sparse environments, improving the interpretability of deep learning surrogates for regulatory acceptance, and fostering tighter integration with digital twin technologies. By embracing these advanced optimization techniques, biomedical researchers can accelerate the pace of innovation, reduce development costs, and ultimately deliver better healthcare solutions.

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