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 highdimensional parameterization of priors. This paper proposes a domain-decomposed variational autoencoder 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 postprocessed 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.

中文翻译：

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贝叶斯反问题的域分解VAE方法

当正演模型由复杂的偏微分方程 (PDE) 控制时，贝叶斯逆问题通常在计算上具有挑战性。这通常是由昂贵的前向模型评估和先验的高维参数化引起的。本文提出了一种域分解变分自编码器马尔可夫链蒙特卡罗（DD-VAE-MCMC）方法来同时应对这些挑战。通过将全局物理域划分为小的子域，该方法首先基于局部历史数据构建局部确定性生成模型，从而提供有效的局部先验表示。具有主动学习功能的高斯过程模型解决了域分解界面条件。然后在低维潜在参数空间中并行地独立地对每个子域进行反演。局部推理解决方案通过泊松图像混合过程进行后处理，以产生有效的全局推理结果。提供了数值示例来证明所提出方法的性能。