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