Bayesian operator inference for data-driven reduced-order modeling

TL;DR

This paper introduces a Bayesian inference approach for data-driven reduced-order modeling of time-dependent systems, enabling uncertainty quantification and hyperparameter selection within the modeling process.

Contribution

It presents a novel Bayesian framework for operator inference in reduced-order models, incorporating uncertainty quantification and hyperparameter optimization.

Findings

Effective uncertainty quantification in reduced-order predictions.

Demonstrated on Euler equations with noisy data and combustion process.

Efficient Monte Carlo sampling of the posterior distribution.

Abstract

This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inverse problem with Gaussian prior and likelihood. The resulting posterior distribution characterizes the operators defining the reduced-order model, hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The statistical moments of these predictions are estimated via a Monte Carlo sampling of the posterior distribution. Since the reduced models are fast to solve, this sampling is computationally efficient. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the regularization term that is present in the deterministic operator inference problem, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated on two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.