(752h) Constructing Efficient Surrogate Models for High-Dimensional First Principles Systems Under Uncertainty

A plethora of chemical engineering processes are informed by sophisticated numerical simulations based on first-principles models (e.g., consisting of a set of coupled partial differential equations) that exhibit highly nonlinear behavior and are computationally expensive. It is often the case that the parameters in these models as well as the initial and boundary conditions in dynamical systems are uncertain and, thus, this uncertainty is propagated to the model outputs. Therefore, relying on deterministic simulation results could possibly lead to erroneous outcomes for the design, operation, and control of these processes. The computational burden associated with solving the entire set of equations can be prohibitive when one wishes to run the model for the whole spectrum of possible input parameter values, even in this era of high-performance computing, especially in the context of optimization applications. Metamodeling techniques aim at alleviating this problem by approximating first-principles model responses under uncertainty with efficient black-box mappings that can serve as surrogate models for the original high-fidelity models [1]. The primary notion in metamodeling is to calibrate the surrogate models using the fewest possible runs of the original model without compromising their predictive capability.

In this work, we explore the effectiveness of various well-established meta-modeling techniques applied to a basic, but relevant crystallization problem, i.e., the growth/dissolution of a 2D crystal population through cooling/heating cycles. This technique is used, possibly in combination with milling, to manipulate the crystal shape of active pharmaceutical ingredients, and efforts have been recently devoted to optimize and control this process. Even though relatively simple, such a problem represents an interesting test-case for metamodeling since it is highly nonlinear, inherently dynamic, and industrially relevant. [2,3] The crystallization system is described by a system of integro-partial differential equations, whose parameters are inherently uncertain. The parameter uncertainties are modeled by suitable probability distributions selected according to available experimental data and expert knowledge.

We first employ sparse polynomial chaos expansions (PCE) whose coefficients are estimated based on least-angle regression and explore their scalability with an increasing number of uncertain inputs. PCEs have been traditionally used for uncertainty quantification in numerous applications (e.g., see [4]). We then present the fundamentals of Gaussian Processes (i.e., Kriging) [5] and explore their coupling with sparse PCEs in order to create surrogate models with optimal local and global predictive capabilities [6]. Finally, the use of NARX neural networks [7] is demonstrated to deal with the continuous dynamic behavior of the system. The accuracy of these three surrogate modeling approaches in capturing the highly nonlinear spatio-temporal dynamics of the crystallization process at hand is assessed using state-of-the-art cross-validation and validation measures, and their computational aspects are discussed.

[1] Bhosekar A., Ierapetritou M., “Advances in surrogate based modeling, feasibility analysis, and optimization: A review”, Computers and Chemical Engineering, 108:250-267, 2018

[2] Salvatori, F., Mazzotti, M., “Manipulation of particle morphology by crystallization, milling, and heating cycles—A mathematical modeling approach”, Industrial and Engineering Chemistry Research, 56:9188–9201, 2017.

[3] Bötschi, S., Rajagopalan, A. K., Morari, M., Mazzotti, M., “Feedback control for the size and shape evolution of needle-like crystals in suspension. I. Concepts and Simulation Studies”, Crystal Growth and Design, 18:4470–4483, 2018.

[4] Paulson J.A., Mesbah A., “An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems”, International Journal of Robust and Nonlinear Control, 1-21, 2017.

[5] Kawai S., Shimoyama K. “Kriging-model-based uncertainty quantification in computational fluid dynamics. 32nd AIAA Applied Aerodynamics Conference, 2014, 2737.

[6] Schöbi R., Sudret, B., Wiart J., “Polynomial-Chaos-based Kriging”, International Journal for Uncertainty Quantification, Begell House Publishers, 5:171-193, 2015.

[7] Mai C.V., Spiridonakos M.D., Chatzi E.N., Sudret B., “Surrogate modeling for stochastic dynamical systems by combining nonlinear autoregressive with exogenous input models and polynomial chaos expansions”, International Journal for Uncertainty Quantification, 6:313-339, 2016.