  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.
