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Solving high-dimensional Bayesian inverse problems (BIPs) with thevariational inference (VI) method is promising but still challenging. The maindifficulties arise from two aspects. First, VI methods approximate theposterior distribution using a simple and analytic variational distribution,which makes it difficult to estimate complex spatially-varying parameters inpractice. Second, VI methods typically rely on gradient-based optimization,which can be computationally expensive or intractable when applied to BIPsinvolving partial differential equations (PDEs). To address these challenges,we propose a novel approximation method for estimating the high-dimensionalposterior distribution. This approach leverages a deep generative model tolearn a prior model capable of generating spatially-varying parameters. Thisenables posterior approximation over the latent variable instead of the complexparameters, thus improving estimation accuracy. Moreover, to accelerategradient computation, we employ a differentiable physics-constrained surrogatemodel to replace the adjoint method. The proposed method can be fullyimplemented in an automatic differentiation manner. Numerical examplesdemonstrate two types of log-permeability estimation for flow in heterogeneousmedia. The results show the validity, accuracy, and high efficiency of theproposed method.

VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems