(46b) A New Method for Building Surrogate Models of Stochastic Simulations

With the increase in computational power and our understanding of fundamental phenomena, we can now develop high-fidelity simulation models to study, design, and optimize engineering systems (Eaton et al., 2017; Ookawara et al., 2007; Szilágyi et al., 2018). Despite these advances, some of these simulation models are still computationally expensive to utilize in sensitivity analysis or optimization studies where many simulation runs may be necessary (Liu et al., 2016; Peherstorfer et al., 2017). Different approaches have been recommended in the literature for handling this computational expense. One commonly-used approach is building a surrogate/meta-model model of the simulation and employing the surrogate model in the analysis when an evaluation of the high-fidelity simulation is required (Han and Zhang, 2012). Surrogate models or meta-models, are representative of the original models, but cheaper to evaluate computationally. The surrogate models approximate the output of the high-fidelity simulation models based on the inputs (Quirante et al., 2015).

When high-fidelity simulation models have uncertain parameters and/or uncertain inputs, the traditional techniques for building surrogate models fail to accurately represent the original model outputs (Staum, 2009). Current approaches for building surrogate models of stochastic simulations, in general, fall into one of two groups. The first group considers a nominal value for the uncertain parameter(s) and builds surrogate models at these values of uncertain parameters (Hüllen et al., 2019) or defines an uncertainty set for the stochastic parameter, evaluates the high-fidelity simulation model using the values of the elements of these sets and carries out the surrogate modeling (Hüllen et al., 2019). The second group, which has been quite popular in recent years, applies stochastic kriging (Ankenman et al., 2008). By utilizing the methods under the first group, more uncertainty may be added to the surrogate model because part of the information is being ignored by fixing the value of uncertain parameters (Hüllen et al., 2019). Stochastic kriging has been shown to exhibit good performance in predicting the expected output value (Ankenman et al., 2008). However, by using this method, the surrogate model type is predetermined because it is built based on kriging meta-models, while other surrogate modeling techniques may be more appropriate for representing the data (Williams and Cremaschi, 2019). Developing approaches for building surrogate models to estimate the outputs of high-fidelity stochastic simulation models accurately can overcome these shortcomings.

In this study, we propose that accurate surrogate models can be built for estimating the outputs of high-fidelity stochastic simulations by considering each uncertain parameter of the stochastic simulation as an uncertain input to the simulation. Isolating the uncertain stochastic parameters of the high-fidelity simulation converts the simulation to a deterministic one, for which most of the popular surrogate modeling techniques can be used to train a meta-model. The uncertain parameter(s) are still part of the meta-model, which paves the way for easier analysis of the uncertainty. Assume we have a high-fidelity simulation model g(X;K), where X is the input vector with dimension d₁, where some of the inputs may be uncertain, and K is the vector of system uncertain parameters with d₂ components. The new simulation model is defined as g'(X*), where X* is a d dimensional vector of inputs with d = d₁+ d₂, which contains all inputs and the uncertain parameters of the stochastic simulation (g(X;K)). It is assumed that the distributions of the uncertain parameters (K) are known, and the parameters of the distributions are constant. With these definitions, the new simulation model (g'(X*)) becomes deterministic, and any of the surrogate modeling techniques can be utilized to train a model representing this simulation (Fig. 1).

With these definitions, the new simulation model (g'(X*)) becomes deterministic, and any of the surrogate modeling techniques can be utilized to train a model representing this simulation.

g'(X*) ≈ F'(X* ), where F'(X* ) is the trained meta-model representing the deterministic formulation. With this new method, the high fidelity stochastic simulation is approximated using the existing surrogate modeling techniques for deterministic models.

There are various machine learning techniques practiced for surrogate modeling. Seven different techniques are implemented in this study to train the meta-models of deterministic formulation, g'(X*). The performance of the new method is evaluated computationally using these seven surrogate modeling methods for test functions with uncertain parameters. The test functions are from the Virtual Library of Simulation Experiments (Surjanovic and Bingham, 2013) with a different number of inputs and uncertain parameters. The quality of the predictions is evaluated for test functions with various features. The number of inputs of the function (dimension of X), number of uncertain parameters (cardinality of K), and nonlinearity of the uncertain parameters are considered as features of the test functions. Root mean square error and the mean absolute error are the metrics of the quality of the performance. One thousand data points generated for each of the test functions were used as the training data set. The test set was 10⁵ new data points for comparison of the model performances. The results suggest that the prediction accuracy depends on the specifications of the test functions. The number of uncertain parameters, as well as the number of input parameters have a significant effect on the quality of the predictions.

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