Surrogate Models

Generating Design Data

In COMSOL Multiphysics^®, data for training surrogate models can be generated from simulation studies, imported from external sources, or created automatically as part of dedicated functionality for UQ and optimization. For general surrogate model training, input parameters and quantities of interest are defined and the parameter space is sampled using DOE methods such as Latin hypercube sampling.

Each sampled input point typically corresponds to a full simulation used to generate the training dataset. To improve efficiency, data generation can be parallelized and distributed across computational resources. Once the data is available, a surrogate model function can be created and trained, providing a quick-evaluating approximation of the simulation results.

For UQ and global optimization studies, specialized surrogate models can be generated using adaptive sampling methods that focus computational effort where it is most valuable.

Surrogate Model Training

Surrogate model training in COMSOL Multiphysics^® is performed directly from the Settingswindow of the corresponding function definition. For deep neural networks (DNNs), the industry-standard Adam solver provides fast and robust training, with hyperparameter settings for learning rate, batch size, number of epochs, validation data, and stopping criteria.

During training, convergence plots show the evolution of the training and validation loss, making it easy to monitor progress and assess model quality.

For Gaussian process (GP) and polynomial chaos expansion (PCE) surrogate models, dedicated adaptive training methods are used to minimize the number of required model solves while building accurate surrogate representations.

DNN Architecture

A DNN surrogate model is built from an input layer, multiple hidden layers, and an output layer, with each layer containing nodes that process information as it moves through the network. In COMSOL Multiphysics^®, the network architecture can be customized by defining the number of layers and the number of nodes per layer. Each layer can also be assigned its own activation function, providing control over how information is transformed at each stage of the network.

Available activation functions include linear, ReLU, ELU, sigmoid, tanh, softplus, leaky ReLU, and GELU.

Geometry Sampling

Geometry sampling enables surrogate models to represent spatially varying fields over a computational domain. Instead of treating spatial coordinates as ordinary design parameters and solving the full model for each sampled point, the model is solved for selected combinations of driving parameters, while field values are stored for the entire geometry or selected parts of it.

This approach makes efficient use of each full solution and can generate large training datasets for DNN surrogate models. Once trained, the surrogate function can quickly reconstruct quantities such as temperature, displacement, stress, or electric potential throughout the geometry while retaining the spatial context of the underlying CAD geometry and mesh.

UQ

Surrogate models based on Gaussian process (GP) and polynomial chaos expansion (PCE) methods are available in the Uncertainty Quantification Module and are central to efficient UQ studies. By approximating the relationship between uncertain input parameters and quantities of interest, they make it possible to perform repeated evaluations at a much lower computational cost than that of running the full simulation model each time.

These surrogate models support UQ studies such as screening, sensitivity analysis, uncertainty propagation, reliability analysis, and inverse UQ. They can help identify the input parameters that most strongly influence the results, quantify their contribution to output variation, evaluate probability distributions of quantities of interest, estimate the risk of exceeding specified thresholds, and infer the distribution of unknown input parameters.

Surrogate Model Functions

Surrogate models are available as general-purpose functions in the COMSOL Multiphysics environment. They provide fast-evaluating approximations of simulation results and can be used to replace an entire physics model or selected parts of a model, accelerate repeated evaluations, support optimization and UQ, and visualize results.

Surrogate models are used for multidimensional function approximation and are well suited for capturing nonlinear behavior in datasets. They provide a flexible alternative to simple linear interpolation, making it possible to represent more complex relationships between inputs and outputs. The functions also support symbolic differentiation with respect to any input parameter, enabling efficient gradient-based analysis and design.

Types of Surrogate Models

The available surrogate model functions include:

Deep Neural Network (DNN) A general-purpose surrogate model that can handle data tables with millions of rows. DNN surrogate models can be trained and used in COMSOL Multiphysics^® without add-on products.

Gaussian Process (GP)¹ A surrogate model that provides uncertainty estimates for the quality of the data fit. GP models are efficient for data tables with up to a few thousand rows.

Polynomial Chaos Expansion (PCE)¹ A surrogate model particularly well suited for UQ and sensitivity analysis. PCE represents quantities of interest using a polynomial basis defined by the probability distributions of the input parameters.

Already-trained surrogate models can be used in COMSOL Multiphysics^® without any add-on products.

Frequency, Time, and Parametric Sampling

DNN surrogate models, as well as specialized surrogate models for UQ, support frequency- and time-dependent quantities of interest. This enables fast evaluation over all computed frequency points, time steps, or parameter values from the underlying study, without requiring additional simulations.

By making use of the full range of computed results, surrogate models can capture how quantities of interest vary with frequency, time, or other parameters. This is especially useful for RF, acoustics, MEMS, and battery simulations, where performance often depends on responses that vary over a frequency or time range. The resulting surrogate functions can be used for visualization, design exploration, optimization, and UQ.

Optimization with Surrogate Models

Surrogate models can be used together with optimization methods in the Optimization Module to reduce the cost of repeated model evaluations. Since surrogate functions evaluate quickly, they are well suited for both local and global optimization.

Local optimization solvers can be applied to an already-trained surrogate model. For global optimization, the efficient global optimization (EGO) solver uses Bayesian optimization to adaptively build a Gaussian process surrogate model of the objective function as the optimization progresses, emphasizing regions where promising objective values are found.