  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.
