Title: Learning physics-based reduced-order models from data: Operator inference for parametric partial differential equations
Abstract: Large-scale numerical simulations of complex physical systems form the backbone of many modern scientific applications. For decades, mathematicians and computational scientists have focused on solving the forward problem of mapping initial/boundary conditions, system parameters, and auxiliary inputs to the corresponding solution of a known dynamical system. Next-generation scientific tasks such as physics-constrained optimization, optimal experimental design, and uncertainty quantification require many forward simulations (sometimes thousands or millions), each with different scenario parameters. Unfortunately, forward problems are often computationally intensive due to spatial and temporal resolution demands. Model order reduction seeks to alleviate the computational burden of forward solves by replacing expensive numerical simulations of complex physical systems with inexpensive surrogate models, called reduced-order models.
Classical model order reduction techniques construct reduced-order models by directly compressing the discretized governing equations, but this approach is infeasible for production-level codes where the discretization details are highly complex, proprietary, or classified. This talk presents Operator Inference, a data-driven model order reduction framework for constructing reduced-order models using only (i) knowledge of the structure of the governing equations and (ii) available simulation data. We detail a method for ensuring stability in the reduced-order model through a regularization selection procedure and use Bayesian inference to quantify the uncertainties associated with the data-driven learning. We also show how, for a large class of parametric systems, parametric dependencies can be embedded directly into the reduced-order model. In this setting, well-posedness conditions for the learning problem lead to a parameter selection criteria. The methodology is demonstrated on a variety of applications, including a single-injector combustion process and the FitzHugh-Nagumo neuron model.