论文标题
贝叶斯操作员推断数据驱动的还原级建模
Bayesian operator inference for data-driven reduced-order modeling
论文作者
论文摘要
这项工作提出了一种贝叶斯推断方法，用于减少时间依赖性系统的阶级建模。在理事方程式的结构中得知，从数据中学习减少订单模型的任务被视为贝叶斯的逆问题，即高斯事先和可能性。所得的后验分布表征了定义还原级模型的运算符，因此降级模型随后发出的预测赋予了不确定性。这些预测的统计矩是通过后验分布的蒙特卡洛采样来估计的。由于还原模型可以快速求解，因此该采样在计算上是有效的。此外，提出的贝叶斯框架提供了确定性操作员推断问题中存在的正则化项的统计解释，最大边缘可能性的经验贝叶斯方法表明，正则化超参数量的选择算法。在两个示例中证明了所提出的方法：可压缩的欧拉方程，具有噪声浪费的观测值和单个注射器燃烧过程。
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.
