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In this paper, we view the statistical inverse problems of partialdifferential equations (PDEs) as PDE-constrained regression and focus onlearning the prediction function of the prior probability measures. From thisperspective, we propose general generalization bounds for learninginfinite-dimensionally defined prior measures in the style of the probabilityapproximately correct Bayesian learning theory. The theoretical framework isrigorously defined on infinite-dimensional separable function space, whichmakes the theories intimately connected to the usual infinite-dimensionalBayesian inverse approach. Inspired by the concept of $\alpha$-differentialprivacy, a generalized condition (containing the usual Gaussian measuresemployed widely in the statistical inverse problems of PDEs) has been proposed,which allows the learned prior measures to depend on the measured data (theprediction function with measured data as input and the prior measure as outputcan be introduced). After illustrating the general theories, the specificsettings of linear and nonlinear problems have been given and can be easilycasted into our general theories to obtain concrete generalization bounds.Based on the obtained generalization bounds, infinite-dimensionallywell-defined practical algorithms are formulated. Finally, numerical examplesof the backward diffusion and Darcy flow problems are provided to demonstratethe potential applications of the proposed approach in learning the predictionfunction of the prior probability measures.

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