# Variational Inverting Network for Statistical Inverse Problems of Partial Differential Equations

To quantify uncertainties of the inverse problems governed by partial differential equations (PDEs), the inverse problems are transformed into statistical inference problems based on Bayes' formula. Recently, infinite-dimensional Bayesian analysis methods are introduced to give a rigorous characterization and construct dimension-independent algorithms. However, there are three major challenges for infinite-dimensional Bayesian methods: prior measures usually only behaves like regularizers (can hardly incorporate prior information efficiently); complex noises (e.g., more practical non-identically distributed noises) are rarely considered; many computationally expensive forward PDEs need to be solved in order to estimate posterior statistical quantities. To address these issues, we propose a general infinite-dimensional inference framework based on a detailed analysis on the infinite-dimensional variational inference method and the ideas of deep generative models that are popular in the machine learning community. 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 named variational inverting network (VINet). This inference framework has the ability to 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 generated in the inference stage in an explicit manner. In numerical experiments, we design specific network structures that yield a computable VINet from the general inference framework.Numerical examples of linear inverse problems governed by an elliptic equation and the Helmholtz equation are given to illustrate the effectiveness of the proposed inference framework.

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