Bayesian operator inference for data-driven reduced-order modeling

This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inversion problem with Gaussian prior and likelihood. The operators defining the reduced-order model, rather than being chosen deterministically, are characterized probabilistically as posterior Gaussian distributions. This embeds uncertainty into the reduced-order model, and hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The learned reduced-order models are computationally efficient, which enables Monte Carlo sampling over the posterior distributions of reduced-order operators. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the Tikhonov regularization incorporated in the operator inference, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated by two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.