12.4 Surrogate modeling

Surrogate modeling is a powerful tool in aerodynamics, allowing engineers to approximate complex systems with simplified mathematical models. These models reduce computational costs and enable efficient analysis and optimization of aircraft designs.

By capturing key input-output relationships, surrogate models provide quick predictions of aerodynamic performance. This allows rapid exploration of design spaces and optimization tasks that would be impractical with full physics-based simulations, accelerating the aircraft development process.

Surrogate modeling fundamentals

• Surrogate models, also known as metamodels, are mathematical approximations of complex physical systems or processes
• They are used to reduce computational costs and enable efficient analysis and optimization in various engineering domains, including aerodynamics
• Surrogate models are particularly valuable when high-fidelity simulations or experiments are time-consuming or expensive to conduct

Definition of surrogate models

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• Surrogate models are simplified representations of a system or process that capture the essential input-output relationships
• They are constructed using a limited number of carefully selected data points from the original model or system
• Surrogate models aim to provide accurate predictions of the system's behavior while being computationally inexpensive to evaluate

Advantages vs physical models

• Surrogate models offer significant computational savings compared to high-fidelity physical models or simulations
• They enable rapid exploration of design spaces and facilitate optimization tasks that would be impractical with the original models
• Surrogate models can provide insights into the underlying relationships between input variables and output responses
• They are particularly useful when the physical models are computationally expensive, have long run times, or require extensive resources

Applications in aerodynamics

• Surrogate models find extensive applications in aerodynamic design and analysis
• They are used to model complex flow phenomena, such as lift, drag, and pressure distributions over aircraft surfaces
• Surrogate models enable rapid evaluation of aerodynamic performance metrics for different design configurations
• They facilitate optimization tasks, such as shape optimization, where the objective is to find the best aerodynamic shape based on specific criteria (lift-to-drag ratio)
• Surrogate models are also employed in uncertainty quantification and sensitivity analysis of aerodynamic systems

Types of surrogate models

• Various types of surrogate models exist, each with its own characteristics, strengths, and limitations
• The choice of surrogate model depends on the nature of the problem, the available data, and the desired accuracy and computational efficiency

Polynomial response surfaces

• Polynomial response surface models approximate the input-output relationships using polynomial functions
• They are constructed by fitting a polynomial equation to the available data points
• Polynomial response surfaces are relatively simple to construct and interpret
• They are suitable for modeling smooth and continuous functions with low to moderate nonlinearity
• Examples include linear regression models and quadratic response surfaces

Kriging models

• Kriging models, also known as Gaussian process regression, are interpolation-based surrogate models
• They assume that the output responses are realizations of a Gaussian process with a specified covariance structure
• Kriging models can capture complex nonlinear relationships and provide uncertainty estimates for the predictions
• They are particularly effective for modeling highly nonlinear and multimodal functions
• Kriging models are widely used in computer experiments and optimization tasks

Radial basis functions

• Radial basis function (RBF) models are interpolation-based surrogate models that use radial basis functions as the building blocks
• RBFs are functions whose values depend on the distance from a center point
• RBF models can capture nonlinear relationships and provide smooth interpolation between data points
• They are suitable for scattered and irregularly spaced data points
• Examples of RBFs include Gaussian, multiquadric, and thin-plate spline functions

Support vector regression

• Support vector regression (SVR) is a machine learning technique used for constructing surrogate models
• SVR aims to find a function that approximates the input-output relationships while minimizing the prediction error
• It employs the concept of support vectors, which are data points that define the margins of the regression function
• SVR can handle nonlinear relationships by using kernel functions to transform the input space
• It is effective in high-dimensional spaces and can handle noisy data

Design of experiments

• Design of experiments (DOE) is a systematic approach to selecting the input points for constructing surrogate models
• DOE aims to efficiently sample the design space to capture the essential relationships between inputs and outputs
• Proper experimental design is crucial for building accurate and reliable surrogate models

Sampling strategies

• Sampling strategies determine how the input points are selected from the design space
• The choice of sampling strategy depends on the characteristics of the problem and the desired properties of the surrogate model
• Common sampling strategies include random sampling, stratified sampling, and space-filling designs
• The goal is to achieve a good coverage of the design space while minimizing the number of sample points

Full factorial designs

• Full factorial designs involve evaluating all possible combinations of the input variables at specified levels
• They provide a comprehensive exploration of the design space and capture all main effects and interactions
• Full factorial designs are suitable for problems with a small number of input variables and levels
• However, they become computationally expensive as the number of variables and levels increases

Fractional factorial designs

• Fractional factorial designs are a subset of full factorial designs that reduce the number of experimental runs
• They are based on the principle of confounding, where certain high-order interactions are assumed to be negligible
• Fractional factorial designs allow for the estimation of main effects and low-order interactions with fewer runs
• They are useful when the number of input variables is large, and the computational budget is limited

Latin hypercube sampling

• Latin hypercube sampling (LHS) is a space-filling design strategy that aims to achieve a uniform coverage of the design space
• In LHS, the range of each input variable is divided into equal intervals, and sample points are selected from each interval
• The sample points are chosen in a way that ensures each interval is represented exactly once for each variable
• LHS provides a good balance between space-filling properties and computational efficiency
• It is widely used in computer experiments and uncertainty quantification studies

Model training and validation

• Model training and validation are essential steps in the development of accurate and reliable surrogate models
• Training involves fitting the surrogate model to the available data points and estimating the model parameters
• Validation assesses the performance of the trained model on independent test data to evaluate its predictive capabilities

Training data requirements

• Sufficient and representative training data is crucial for building accurate surrogate models
• The training data should cover the relevant regions of the design space and capture the important relationships between inputs and outputs
• The number of training points depends on the complexity of the problem, the dimensionality of the input space, and the chosen surrogate model type
• Adequate sampling techniques, such as design of experiments, should be employed to generate the training data

Model fitting techniques

• Model fitting techniques are used to estimate the parameters of the surrogate model based on the training data
• The choice of fitting technique depends on the type of surrogate model and the nature of the problem
• Common fitting techniques include least squares regression, maximum likelihood estimation, and gradient-based optimization
• The goal is to minimize the discrepancy between the surrogate model predictions and the actual output values

Cross-validation methods

• Cross-validation is a technique used to assess the predictive performance of a surrogate model
• It involves dividing the available data into training and validation subsets
• The model is trained on the training subset and evaluated on the validation subset
• Common cross-validation methods include k-fold cross-validation and leave-one-out cross-validation
• Cross-validation helps to detect overfitting and provides an estimate of the model's generalization error

Assessing model accuracy

• Assessing the accuracy of a surrogate model is crucial to ensure its reliability and usefulness
• Various metrics can be used to quantify the model's predictive performance, such as mean squared error (MSE), root mean squared error (RMSE), and coefficient of determination (R-squared)
• The choice of metric depends on the specific problem and the goals of the analysis
• Visual inspection of the model's predictions versus the actual values can provide insights into the model's behavior and identify any systematic biases or errors

Surrogate model optimization

• Surrogate model optimization involves using the surrogate model as a substitute for the original expensive model in optimization tasks
• The goal is to find the optimal design or solution by efficiently exploring the design space using the surrogate model

Optimization algorithms

• Various optimization algorithms can be employed in conjunction with surrogate models
• The choice of algorithm depends on the characteristics of the problem, such as the presence of constraints, the nature of the objective function, and the dimensionality of the search space
• Common optimization algorithms include gradient-based methods (steepest descent, conjugate gradient), evolutionary algorithms (genetic algorithms, particle swarm optimization), and pattern search methods

Constraints handling

• Many optimization problems involve constraints that limit the feasible region of the design space
• Surrogate models need to incorporate constraint handling techniques to ensure that the optimal solutions satisfy the specified constraints
• Constraint handling methods include penalty functions, barrier methods, and constrained optimization algorithms (sequential quadratic programming)
• The surrogate model can be used to approximate both the objective function and the constraint functions

Multi-objective optimization

• Multi-objective optimization involves optimizing multiple conflicting objectives simultaneously
• Surrogate models can be employed to approximate the Pareto front, which represents the set of optimal trade-off solutions
• Techniques such as weighted sum methods, epsilon-constraint methods, and evolutionary algorithms can be used to solve multi-objective optimization problems with surrogate models
• The surrogate models can be used to evaluate the objectives and guide the search process towards the Pareto front

Robust design optimization

• Robust design optimization aims to find designs that are insensitive to uncertainties or variations in the input parameters
• Surrogate models can be used to quantify the robustness of a design by incorporating uncertainty propagation techniques
• Robust optimization methods, such as robust counterpart optimization and reliability-based design optimization, can be applied with surrogate models
• The goal is to find designs that maintain good performance under varying conditions or uncertainties

Surrogate-based sensitivity analysis

• Sensitivity analysis aims to identify the input variables that have the most significant impact on the output responses
• Surrogate models can be used to perform sensitivity analysis efficiently by approximating the relationships between inputs and outputs

Local sensitivity analysis

• Local sensitivity analysis evaluates the sensitivity of the output to small perturbations in the input variables around a specific point
• It provides information about the local behavior of the model and the relative importance of the input variables
• Local sensitivity measures include partial derivatives, finite differences, and one-at-a-time (OAT) methods
• Surrogate models can be used to estimate the local sensitivities by evaluating the model at perturbed input points

Global sensitivity analysis

• Global sensitivity analysis considers the entire range of input variables and assesses their overall influence on the output
• It provides a comprehensive understanding of the model's behavior and the interactions between input variables
• Global sensitivity methods include variance-based methods (Sobol' indices), screening methods (Morris method), and regression-based methods
• Surrogate models enable efficient computation of global sensitivity measures by sampling the input space and evaluating the model at multiple points

Variance-based methods

• Variance-based methods decompose the output variance into contributions from individual input variables and their interactions
• Sobol' indices are a popular variance-based sensitivity measure that quantifies the main effects and total effects of input variables
• Surrogate models can be used to estimate the Sobol' indices by evaluating the model at carefully selected input points
• Variance-based methods provide a global assessment of variable importance and help identify the most influential variables

Derivative-based methods

• Derivative-based methods assess the sensitivity of the output to changes in the input variables using partial derivatives
• They provide local sensitivity information and are suitable for smooth and differentiable models
• Adjoint methods and automatic differentiation techniques can be used to compute the derivatives efficiently
• Surrogate models can be differentiated analytically or numerically to obtain the sensitivity measures
• Derivative-based methods are useful for optimization and gradient-based sensitivity analysis

Uncertainty quantification

• Uncertainty quantification (UQ) aims to characterize and quantify the uncertainties associated with the model inputs, parameters, and outputs
• Surrogate models play a crucial role in UQ by enabling efficient propagation of uncertainties and estimation of output statistics

Sources of uncertainty

• Uncertainties can arise from various sources, such as measurement errors, model assumptions, and inherent variability in the system
• Input parameter uncertainty refers to the uncertainty in the values of the input variables
• Model form uncertainty arises from the simplifications and assumptions made in the modeling process
• Numerical uncertainty is associated with the discretization and solution methods used in the simulations

Propagation of uncertainty

• Uncertainty propagation involves determining how the input uncertainties propagate through the model and affect the output quantities of interest
• Monte Carlo simulation is a common approach for uncertainty propagation, where the model is evaluated multiple times with randomly sampled input values
• Surrogate models can be used to accelerate the Monte Carlo simulations by providing fast approximations of the model outputs
• Other uncertainty propagation methods include polynomial chaos expansion and stochastic collocation

Surrogate-based UQ methods

• Surrogate-based UQ methods leverage the computational efficiency of surrogate models to perform uncertainty quantification tasks
• Polynomial chaos expansion (PCE) is a popular surrogate-based UQ method that represents the model output as a series expansion of orthogonal polynomials
• PCE coefficients can be estimated using regression or projection techniques based on a limited number of model evaluations
• Stochastic collocation methods construct surrogate models by evaluating the model at specific collocation points and interpolating the results
• Surrogate-based UQ methods enable efficient estimation of output statistics, such as moments and probability density functions

Reliability analysis

• Reliability analysis assesses the probability of a system or component meeting its performance requirements under uncertain conditions
• Surrogate models can be employed to perform reliability analysis by approximating the limit state function, which separates the safe and failure regions
• Reliability methods, such as first-order reliability method (FORM) and second-order reliability method (SORM), can be applied with surrogate models
• The surrogate models enable efficient estimation of the reliability index and the probability of failure
• Reliability-based design optimization (RBDO) integrates reliability analysis with optimization to find designs that meet reliability targets

Multifidelity surrogate modeling

• Multifidelity surrogate modeling involves combining models of different fidelities to improve the accuracy and efficiency of the surrogate model
• It leverages the information from low-fidelity models, which are computationally cheap but less accurate, and high-fidelity models, which are expensive but more accurate

Hierarchical surrogate models

• Hierarchical surrogate models are constructed by building a hierarchy of models with increasing fidelity levels
• The low-fidelity models are used to capture the global trends and provide a coarse approximation of the system behavior
• The high-fidelity models are used to refine the predictions in the regions of interest or where higher accuracy is required
• The hierarchical structure allows for efficient allocation of computational resources and improves the overall accuracy of the surrogate model

Cokriging methods

• Cokriging is a multifidelity surrogate modeling technique that extends the kriging method to incorporate multiple levels of fidelity
• It exploits the correlation between the low-fidelity and high-fidelity models to improve the prediction accuracy
• Cokriging models the discrepancy between the fidelity levels using a Gaussian process and estimates the model parameters based on the available data
• The cokriging surrogate model can provide accurate predictions by leveraging the information from both low-fidelity and high-fidelity models

Fusion of variable-fidelity data

• Fusion methods aim to combine variable-fidelity data from different sources or experiments to construct a unified surrogate model
• They address the challenges of inconsistencies, noise, and varying levels of fidelity in the available data
• Bayesian model averaging and Bayesian hierarchical modeling are popular fusion techniques that assign weights to different models based on their fidelity and uncertainty
• Fusion methods enable the integration of heterogeneous data sources and provide a coherent representation of the system behavior

Multifidelity optimization strategies

• Multifidelity optimization strategies leverage surrogate models of different fidelities to accelerate the optimization process
• They aim to balance the computational cost and accuracy by adaptively selecting the appropriate fidelity level at each optimization iteration
• Trust-region methods and adaptive sampling techniques are commonly used in multifidelity optimization
• The optimization algorithm starts with the low-fidelity model and progressively refines the solution using higher-fidelity models in the promising regions
• Multifidelity optimization strategies can significantly reduce the computational burden while maintaining the desired level of accuracy

Surrogate model management

• Surrogate model management involves the efficient utilization and adaptation of surrogate models throughout the analysis and optimization process
• It addresses the challenges of model selection, updating, and refinement as new data becomes available or the system evolves

Adaptive sampling techniques

• Adaptive sampling techniques aim to iteratively select the most informative sample points to improve the accuracy of the surrogate model
• They balance the exploration of the design space and the exploitation of the regions with high uncertainty or potential for improvement
• Common adaptive sampling strategies include expected improvement, maximum variance, and maximum entropy
• Adaptive sampling allows for efficient allocation of computational resources and progressive refinement of the surrogate model

Model updating and refinement

• As new data becomes available or the system undergoes changes, the surrogate model needs to be updated and refined to maintain its accuracy
• Model updating techniques incorporate the new information into the existing surrogate model without completely rebuilding it
• Incremental learning methods, such as online kriging and recursive least squares, can be used to update the model parameters efficiently
• Model refinement strategies focus on improving the model's accuracy in specific regions of interest or where the model's predictions are less reliable

Exploitation vs exploration

• Surrogate model management involves balancing the exploitation of the current model's knowledge and the exploration of new regions in the design space
• Exploitation refers to using the surrogate model to identify promising solutions or regions based on the current knowledge
• Exploration involves sampling new points in unexplored or uncertain regions to improve the model's coverage and reduce the risk of missing important features
• Techniques such as expected improvement and upper confidence bound can be used to guide the trade-off between exploitation and exploration

Convergence criteria