Model Structural Inference using Local Dynamic Operators

TL;DR

This paper introduces a novel approach for quantifying model-structure uncertainty in dynamical systems by defining local dynamical operators, enabling non-intrusive model identification, reduced order modeling, and Bayesian calibration.

Contribution

It proposes a new local dynamical operator framework for non-intrusive model structure inference and uncertainty quantification in PDE-based systems.

Findings

Successful identification of model structure from numerical code output

Effective reduced order modeling of complex PDE systems

Calibration of model parameters using Bayesian inference

Abstract

This paper focuses on the problem of quantifying the effects of model-structure uncertainty in the context of time-evolving dynamical systems. This is motivated by multi-model uncertainty in computer physics simulations: developers often make different modeling choices in numerical approximations and process simplifications, leading to different numerical codes that ostensibly represent the same underlying dynamics. We consider model-structure inference as a two-step methodology: the first step is to perform system identification on numerical codes for which it is possible to observe the full state; the second step is structural uncertainty quantification (UQ), in which the goal is to search candidate models "close" to the numerical code surrogates for those that best match a quantity-of-interest (QOI) from some empirical dataset. Specifically, we: (1) define a discrete, local representation of the structure of a partial differential equation, which we refer to as the "local dynamical operator" (LDO); (2) identify model structure non-intrusively from numerical code output; (3) non-intrusively construct a reduced order model (ROM) of the numerical model through POD-DEIM-Galerkin projection; (4) perturb the ROM dynamics to approximate the behavior of alternate model structures; and (5) apply Bayesian inference and energy conservation laws to calibrate a LDO to a given QOI. We demonstrate these techniques using the two-dimensional rotating shallow water (RSW) equations as an example system.