Learning prediction function of prior measures for statistical inverse problems of partial differential equations

In this paper, we view the statistical inverse problems of partial differential equations (PDEs) as PDE-constrained regression and focus on learning the prediction function of the prior probability measures. From this perspective, we propose general generalization bounds for learning infinite-dimensionally defined prior measures in the style of the probability approximately correct Bayesian learning theory. The theoretical framework is rigorously defined on infinite-dimensional separable function space, which makes the theories intimately connected to the usual infinite-dimensional Bayesian inverse approach. Inspired by the concept of $\alpha$-differential privacy, a generalized condition (containing the usual Gaussian measures employed widely in the statistical inverse problems of PDEs) has been proposed, which allows the learned prior measures to depend on the measured data (the prediction function with measured data as input and the prior measure as output can be introduced). After illustrating the general theories, the specific settings of linear and nonlinear problems have been given and can be easily casted into our general theories to obtain concrete generalization bounds. Based on the obtained generalization bounds, infinite-dimensionally well-defined practical algorithms are formulated. Finally, numerical examples of the backward diffusion and Darcy flow problems are provided to demonstrate the potential applications of the proposed approach in learning the prediction function of the prior probability measures.