A domain-decomposed VAE method for Bayesian inverse problems

The forward model, usually defined through partial differential equations (PDEs), describes certain physical phenomena with parameters as inputs. Generally, solving the forward model is computationally expensive but well-defined. In contrast, the related inverse problem which aims at inferring hidden parameters that cannot be directly observed from limited and noisy observations is typically ill-posed: different sets of parameters can result in similar sensor measurements, and there may be no feasible solution to fit the observed data, or minor errors can render unpredictable changes in the forward model. The Bayesian methods [5, 6], by viewing the unknown parameters as random variables, formulate the inverse problem into a probabilistic problem to capture the uncertainty in observations, forward models, and prior knowledge. One can assign a prior distribution to reflect our knowledge of the parameters before any measurements are made. The likelihood function is characterized through the forward model.