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.

翻译：本文将偏微分方程统计逆问题视为PDE约束回归，并聚焦于学习先验概率测度的预测函数。基于此视角，我们提出了针对无限维定义先验测度学习的通用泛化界，该研究遵循概率近似正确贝叶斯学习理论框架。该理论框架在无限维可分函数空间上严格定义，使得相关理论与常规无限维贝叶斯逆方法紧密关联。受$\alpha$-差分隐私概念启发，我们提出了一种广义条件（涵盖PDE统计逆问题中广泛采用的高斯测度），该条件允许学习得到的先验测度依赖测量数据（可引入以测量数据为输入、先验测度为输出的预测函数）。在阐述通用理论后，我们给出了线性和非线性问题的具体设置，这些设置可方便地纳入通用理论框架以获取具体泛化界。基于所获泛化界，我们制定了无限维适定的实用算法。最后，通过反扩散和达西流问题的数值算例，展示了该方法在学习先验概率测度预测函数中的潜在应用价值。