  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.
