Nonparametric Prior Learning in Differential Equation Modeling
提出一种从历史数据中学习先验分布预测函数的框架，用于偏微分方程约束的非参数回归，并给出泛化误差估计和数值验证。
This article addresses Bayesian inference related to partial differential equations (PDEs), particularly nonparametric regression constrained by PDEs. To effectively encode prior information, we propose a novel framework that learns a prediction function of the prior distribution from historical training datasets. We introduce hyper-prior and hyper-posterior distributions and derive a generalization error estimate, which accommodates data-dependent priors by extending the concept of differential privacy. Some mild conditions are given to validate the error estimate, where various typical PDEs such as diffusion and Darcy flow equations can be integrated. We thus formulate an infinite-dimensional optimization problem to obtain the point estimate of the hyper-posterior. Numerical examples demonstrate the performance of our proposed method in learning the prediction function of priors. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.