(284c) Accurate Surrogate Models for Stochastic Simulations

With the progress in understanding the fundamentals of the phenomena and increased computational power, high-fidelity simulation models became widespread to study, design, and optimize engineering systems in recent years (Eaton et al., 2017; Szilágyi et al., 2018). However, in many cases, the complexity of the models requires computational power beyond what is available for applications like sensitivity analysis or optimization studies (Liu et al., 2016; Peherstorfer et al., 2017). These applications require a high number of simulation runs, which leads to high computational costs. One of the common ways to reduce the computational costs is to use the surrogate/meta models built based on the simulation (Han and Zhang, 2012). Surrogate models or meta-models use input-output data from the simulation models for approximating the relationship (Quirante et al., 2015). A vast number of techniques have been successfully used for building surrogate models (e.g.,(Breiman, 2001; Friedman, 1991; Haleem et al., 2013; Williams and Rasmussen, 2006)).

The surrogate models built using traditional techniques do not accurately represent the outputs of high-fidelity stochastic simulations, e.g., simulations with uncertain parameters (Staum, 2009). Current literature for modeling the outputs of a stochastic simulation can be grouped into two categories. In the first category, the uncertain parameter(s) are fixed at a select subset of their values, and a surrogate model is trained for each value of the uncertain parameter (Hüllen et al., 2019). In some cases, the subset may include only one nominal value. However, part of the uncertainty information is lost while using this approach because of the fixed parameter values. The second category employs stochastic kriging to construct the surrogate model (Ankenman et al., 2008). Although stochastic kriging has yielded promising results in predicting the expected output results (Ankenman et al., 2008), the surrogate model technique is fixed. The comparative analysis of different surrogate modeling techniques reveals that the best surrogate modeling technique depends on the input-output data characteristics (Williams and Cremaschi, 2021, 2019).

This work introduces a new approach for building accurate surrogate models of stochastic simulations and compares its performance to the existing approaches. Our approach considers the uncertain parameter(s) as uncertain input(s) to the simulations. By this approach, the high-fidelity stochastic simulation is converted to a deterministic one. The most appropriate surrogate modeling technique is used to approximate the output of the deterministic model accurately. Next, the impact of uncertain parameters is propagated to the outputs using the surrogate model and an efficient uncertainty propagation method (Mohammadi and Cremaschi, 2019), yielding the stochastic simulation output approximation.

The proposed approach is illustrated in Figure 1. Let Y=g(X;K) be a high-fidelity simulation model, where Y is the stochastic output, X is the input vector with dimension d_1, and K is the vector of system uncertain parameters with d₂ components. The new simulation model is defined as Y^′=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)). We assume 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 (Y^′=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 surrogate model of the deterministic simulation (Figure 1).

Seven different machine learning techniques are studied to build the surrogate model, F^′(X^*), for the deterministic model, g^′(X^*). The performance of the new method is evaluated computationally for a set of test functions with uncertain parameters. The test functions are chosen from the Virtual Library of Simulation Experiments (Surjanovic and Bingham, 2013) with different numbers of inputs and uncertain parameters to investigate their impact. We also compare the performance of the new method and the existing approaches in the literature. In the first approach, the uncertain parameter(s) are fixed at their nominal values, which is considered the base case. The second approach considers a subset of fixed values for the uncertain parameter(s), and the third one is stochastic kriging. The quality of the estimates is evaluated based on the number of inputs of the function (dimension of X), number of uncertain parameters (cardinality of K), and nonlinearity of the uncertain parameters as features of the test functions. The metrics for evaluation of the performance are the root mean square error and the mean absolute error calculated for the predicted outputs, which are Yˆ and the standard deviation associated with it. This presentation will discuss the improvements in performance metrics of all the methods compared to the base case.

References

Ankenman, B., Nelson, B.L., Staum, J., 2008. Stochastic kriging for simulation metamodeling. Proc. - Winter Simul. Conf. 362–370. https://doi.org/10.1109/WSC.2008.4736089

Breiman, L., 2001. Random forests. Mach. Learn. 45, 5–32.

Eaton, A.N., Beal, L.D.R., Thorpe, S.D., Hubbell, C.B., Hedengren, J.D., Nybø, R., Aghito, M., 2017. Real time model identification using multi-fidelity models in managed pressure drilling. Comput. Chem. Eng. 97, 76–84. https://doi.org/10.1016/j.compchemeng.2016.11.008

Friedman, J.H. (stanford U., 1991. Multivariate adaptive regression splines.

Haleem, K., Gan, A., Lu, J., 2013. Using multivariate adaptive regression splines ( MARS ) to develop crash modification factors for urban freeway interchange influence areas. Accid. Anal. Prev. 55, 12–21. https://doi.org/10.1016/j.aap.2013.02.018

Han, Z.-H., Zhang, K.-S., 2012. Surrogate-based optimization. Real-world Appl. Genet. algorithms 343–362.

Hüllen, G., Zhai, J., Kim, S.H., Sinha, A., Realff, M.J., Boukouvala, F., 2019. Managing Uncertainty in Data-Driven Simulation-Based Optimization. Comput. Chem. Eng. 106519. https://doi.org/10.1016/j.compchemeng.2019.106519

Liu, B., Koziel, S., Zhang, Q., 2016. A multi-fidelity surrogate-model-assisted evolutionary algorithm for computationally expensive optimization problems. J. Comput. Sci. 12, 28–37. https://doi.org/10.1016/j.jocs.2015.11.004

Mohammadi, S., Cremaschi, S., 2019. Efficiency of Uncertainty Propagation Methods for Estimating Output Moments, in: Muñoz, S.G., Laird, C.D., Realff, M.J. (Eds.), Proceedings of the 9th International Conference on Foundations of Computer-Aided Process Design, Computer Aided Chemical Engineering. Elsevier, pp. 487–492. https://doi.org/https://doi.org/10.1016/B978-0-12-818597-1.50078-3

Peherstorfer, B., Kramer, B., Willcox, K., 2017. Combining multiple surrogate models to accelerate failure probability estimation with expensive high-fidelity models. J. Comput. Phys. 341, 61–75. https://doi.org/10.1016/j.jcp.2017.04.012

Quirante, N., Javaloyes, J., Ruiz-Femenia, R., Caballero, J.A., 2015. Optimization of chemical processes using surrogate models based on a kriging interpolation, in: Computer Aided Chemical Engineering. Elsevier, pp. 179–184.

Staum, J., 2009. Better Simulation Metamodeling: The why, what, and how of Stochastic Kriging 119–133.

Surjanovic, S., Bingham, D., 2013. Virtual Library of Simulation Experiments: Test Functions and Datasets.

Szilágyi, B., Agachi, P.Ş., Nagy, Z.K., 2018. Chord Length Distribution Based Modeling and Adaptive Model Predictive Control of Batch Crystallization Processes Using High Fidelity Full Population Balance Models. Ind. Eng. Chem. Res. 57, 3320–3332. https://doi.org/10.1021/acs.iecr.7b03964

Williams, B., Cremaschi, S., 2021. Selection of Surrogate Modeling Techniques for Surface Approximation and Surrogate-Based Optimization. Chem. Eng. Res. Des. https://doi.org/https://doi.org/10.1016/j.cherd.2021.03.028

Williams, B.A., Cremaschi, S., 2019. Surrogate Model Selection for Design Space Approximation And Surrogatebased Optimization, in: Computer Aided Chemical Engineering. Elsevier, pp. 353–358.

Williams, C.K.I., Rasmussen, C.E., 2006. Gaussian processes for machine learning. MIT press Cambridge, MA.