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.

这项工作提出了一种贝叶斯推理方法，用于时间相关系统的降阶建模。根据控制方程的结构，从数据中学习降阶模型的任务被提出为具有高斯先验和似然性的贝叶斯逆问题。由此产生的后验分布表征了定义降阶模型的算子，因此降阶模型随后发布的预测具有不确定性。这些预测的统计矩是通过后验分布的蒙特卡罗抽样来估计的。由于简化模型可以快速求解，因此这种采样在计算上是有效的。此外，所提出的贝叶斯框架提供了确定性算子推理问题中存在的正则化项的统计解释，最大边际似然的经验贝叶斯方法提出了正则化超参数的选择算法。所提出的方法在两个示例上进行了演示：具有噪声破坏观测的可压缩欧拉方程和单喷射器燃烧过程。