To quantify uncertainties in inverse problems of partial differential equations (PDEs), we formulate them into statistical inference problems using Bayes' formula. Recently, well-justified infinite-dimensional Bayesian analysis methods have been developed to construct dimension-independent algorithms. However, there are three challenges for these infinite-dimensional Bayesian methods: prior measures usually act as regularizers and are not able to incorporate prior information efficiently; complex noises, such as more practical non-i.i.d. distributed noises, are rarely considered; and time-consuming forward PDE solvers are needed to estimate posterior statistical quantities. To address these issues, an infinite-dimensional inference framework has been proposed based on the infinite-dimensional variational inference method and deep generative models. Specifically, by introducing some measure equivalence assumptions, we derive the evidence lower bound in the infinite-dimensional setting and provide possible parametric strategies that yield a general inference framework called the Variational Inverting Network (VINet). This inference framework can encode prior and noise information from learning examples. In addition, relying on the power of deep neural networks, the posterior mean and variance can be efficiently and explicitly generated in the inference stage. In numerical experiments, we design specific network structures that yield a computable VINet from the general inference framework. Numerical examples of linear inverse problems of an elliptic equation and the Helmholtz equation are presented to illustrate the effectiveness of the proposed inference framework.

翻译：为量化偏微分方程反问题中的不确定性，我们利用贝叶斯公式将其构建为统计推断问题。近年来，已发展出具有严格理论基础的无限维贝叶斯分析方法以构建维度无关算法。然而，这些无限维贝叶斯方法存在三个挑战：先验测度通常仅作为正则化项，无法有效融入先验信息；复杂噪声（如更具实用性的非独立同分布噪声）很少被考虑；估计后验统计量需要耗费大量时间求解正问题偏微分方程。为解决上述问题，本文基于无限维变分推断方法与深度生成模型提出了一种无限维推断框架。具体而言，通过引入若干测度等价假设，我们推导出无限维情形下的证据下界，并提供可行的参数化策略，从而构建出名为变分逆网络（VINet）的通用推断框架。该推断框架能从学习样例中编码先验与噪声信息。此外，依托深度神经网络的强大能力，后验均值与方差可在推断阶段高效显式生成。在数值实验中，我们设计了特定网络结构，从通用推断框架中得到可计算的VINet。通过椭圆方程与亥姆霍兹方程的线性反问题数值算例，验证了所提推断框架的有效性。