Bayesian inverse problems are often computationally challenging when the
forward model is governed by complex partial differential equations (PDEs).
This is typically caused by expensive forward model evaluations and
high-dimensional parameterization of priors. This paper proposes a
domain-decomposed variational auto-encoder Markov chain Monte Carlo
(DD-VAE-MCMC) method to tackle these challenges simultaneously. Through
partitioning the global physical domain into small subdomains, the proposed
method first constructs local deterministic generative models based on local
historical data, which provide efficient local prior representations. Gaussian
process models with active learning address the domain decomposition interface
conditions. Then inversions are conducted on each subdomain independently in
parallel and in low-dimensional latent parameter spaces. The local inference
solutions are post-processed through the Poisson image blending procedure to
result in an efficient global inference result. Numerical examples are provided
to demonstrate the performance of the proposed method.