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.

翻译：贝叶斯逆问题在前向模型受复杂偏微分方程（PDE）支配时，通常面临计算挑战，这往往源于昂贵的正向模型评估及先验的高维参数化。本文提出一种域分解变分自编码器马尔可夫链蒙特卡洛（DD-VAE-MCMC）方法，以同时应对这些挑战。通过将全局物理域划分为多个子域，该方法首先基于局部历史数据构建局部确定性生成模型，从而提供高效的局部先验表示。采用带有主动学习的高斯过程模型处理域分解界面条件。随后，在各子域上独立并行进行低维潜在参数空间中的反演。通过泊松图像融合技术对局部推断解进行后处理，最终获得高效的全局推断结果。文中提供数值算例以验证所提方法的性能。