Computer modeling is designed to predict the behavior of complex systems by solving the corresponding mathematical equations of the physical process. The simulation requires a huge number of parameters to be found, and they all require a large number of simulation runs, where each run takes a diferent combination of design parameters as input. Computer simulations tend to be expensive. Research that requires a large number of computer calculations will lead to exorbitant computational costs, which will make them practically impracticable. This is what surrogate modeling is for. Surrogate modeling builds a statistical model to accurately approximate the simulation result. Subsequently, this trained model can replace the original computer simulation when performing system analysis.
Recently, statistical modeling, a subsection of mathematical modeling that uses the mathematical
apparatus of machine learning [
Surrogate modeling builds a statistical model to accurately approximate the simulation result. Subsequently, this trained model can replace the original computer simulation when performing system analysis. Surrogate modeling methods make a resource-intensive analysis Workshop on information technology and scientific computing in the framework of the XI International Conference LGOBE
CEUR of a computational model more accessible, since evaluating the trained statistical model is much faster than evaluating the original simulation, performing many experiments with various combinations of design parameters.
The process of building a surrogate model can be divided into three main stages, which can be alternated iteratively: • Receiving and preprocessing data. As data can be used as data obtained as a result of tests and computational experiments or an analytical method. In addition, this stage involves reducing the dimension of the data by removing unnecessary or redundant parameters from the dataset and forming a training sample. • Construction of a surrogate model and optimization of model parameters. At this stage, the method for implementing the model should be chosen. The most common methods include models such as Kriging, Artificial Neural Networks, Support Vector Regression, Multivariate Nonparametric Regression, Polynomial Regression, etc. The model allows constructing an approximation based on the selected training sample. For best results, you can plot multiple approximation curves. • Estimating the accuracy of the surrogate model. The accuracy of the surrogate model depends on the quantity and quality of the data. In addition, a poor fit can arise from noise in the data or from poorly chosen surrogate model building techniques.
The first step in building a surrogate model is collecting training data. After collecting the input data and the corresponding output data into a training set, we can build a statistical model based on this set.
Let’s list the main methods of data selection when building surrogate models: • random selection; • Latin sample of the hypercube; • factorial sampling; • low discrepancy sequences.
After collecting the data, we build the model. Now there are many surrogate models, and the choice of model depends on the task: classification, approximation, forecast, etc.
Let’s list the main methods for constructing and training surrogate models: • linear models; • radially basic models; • kriging; • inverse distance weighted method; • spline method; • 2nd order polynomial method.
Let’s list the most popular general-purpose surrogate modeling packages.
• Matlab toolbox SUrrogate MOdeling (SUMO) [
SciML/Surrogates.jl package is for surrogate modeling and optimization for scientific machine learning. To install the package, just run the commands as follows: using Pkg Pkg.add("Surrogates")
The SciML/Surrogates.jl package provides all the necessary tools for the complete process of creating a surrogate model. We will build a surrogate model in three stages. • Sample selection. The following methods are implemented for sampling: – Grid sample; – Uniform sample; – Sobol sample; – Latin Hypercube sample; – Low discrepancy sample. – Linear; – Radial Basis; – Kriging; – Neural Network; – Support Vector Machine; – Random Forest; – Second Order Polynomial; – Inverse Distance. • Building a surrogate model. The following surrogates are available: • Optimization. After the surrogate is built, it needs to be optimized for some objective function. The following optimization methods are available: – Stochastic RBF (SRBF); – Lower confidence bound strategy (LCBS); – Expected improvement (EI); – Dynamic coordinate search (DYCORS).
Surrogate Modeling Toolkit (SMT) [
SMT consists of three main modules that implement a set of sampling methods, benchmarking functions, and surrogate modeling methods, respectively. Each module contains a common interface inherited by the corresponding methods, and each method implements the functionality required by the interface.
There are two main steps required to build a surrogate model. First, we generate a set of training points from the input space. This step can be done using one of the sampling methods implemented in SMT, or by loading an existing training dataset. Second, we train the desired surrogate model at these points and make a prediction of the outputs and derivatives.
SMT contains a library of sampling techniques used to generate sets of points in the input space for both training and prediction.
Let’s consider the construction of surrogates using the SciML/Surrogates.jl library as an example.
Let us consider various methods using the example of collecting samples to build a surrogate model for the simplest function as follows () = sin .
(1)
To generate samples by various methods for the function under consideration on the interval [0, 30] at 15 points using the built-in functions of the SciML/Surrogates.jl package: # Grid method x_grid = sample(n_samples, lower_bound, upper_bound, GridSample(0.2)) y_grid = f.(x_grid) # Uniform method x_uniform = sample(n_samples, lower_bound, upper_bound, UniformSample()) y_uniform = f.(x_uniform) # Sobol method x_sobol = sample(n_samples, lower_bound, upper_bound, SobolSample()) y_sobol = f.(x_sobol) # Latin hypercube method x_lh = sample(n_samples, lower_bound, upper_bound, LatinHypercubeSample()) y_lh = f.(x_lh)
The graphic 1 shows the results of generating samples. It can be seen from the graph that for this example, the best examples are provided by the Sobol sequence methods and the Latin hypercube method.
To construct a surrogate model of the (1) function, we use various methods implemented in the SciML/Surrogates.jl library: # Radial basis method RBF_surrogate = RadialBasis(x, y, lower_bound, upper_bound) # Kriging kriging_surrogate = Kriging(x, y, lower_bound, upper_bound, p=1.9) # Lobachesky method lobachevsky_surrogate = LobacheskySurrogate(x, y, lower_bound, upper_bound, ↪ alpha = 2.0, n = 6) # Inverse distance method InverseDistance = InverseDistanceSurrogate(x, y, lower_bound, upper_bound)
The simulation results using these methods for the sequences obtained by the Sobol selection method and the Latin Hypercube selection method are presented in the graphics 2 and 3.
It can be concluded that for the studied example for constructing a surrogate model, the radially basic method gives the worst of the presented results. While kriging and Lobachevsky’s method give approximately the same picture.
In this article, we reviewed the main opensource surrogate modeling toolkits. To study the features of software implementations, a simple surrogate model was built. The Surrogates.jl and SMT libraries are given as examples of surrogate modeling implementation. From the point of view of the authors, the Python SMT library is currently slightly superior to the Julia library Surrogates.jl.
This paper has been supported by the RUDN University Strategic Academic Leadership Program (Anastasia V. Demidova), by the Russian Science Foundation grant No. 19-71-30008 (Tatyana R. Velieva) by the Russian Foundation for Basic Research (RFBR) according to the research project No 19-01-00645 (Anna V. Korolkova, Dmitry S. Kulyabov).