Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems

Estimation of spatially-varying parameters for computationally expensive forward models governed by partial differential equations is addressed. A novel multiscale Bayesian inference approach is introduced based on deep probabilistic generative models. Such generative models provide a flexible representation by inferring on each scale a low-dimensional latent encoding while allowing hierarchical parameter generation from coarse- to fine-scales. Combining the multiscale generative model with Markov Chain Monte Carlo (MCMC), inference across scales is achieved enabling us to efficiently obtain posterior parameter samples at various scales. The estimation of coarse-scale parameters using a low-dimensional latent embedding captures global and notable parameter features using an inexpensive but inaccurate solver. MCMC sampling of the fine-scale parameters is enabled by utilizing the posterior information in the immediate coarser-scale. In this way, the global features are identified in the coarse-scale with inference of low-dimensional variables and inexpensive forward computation, and the local features are refined and corrected in the fine-scale. The developed method is demonstrated with two types of permeability estimation for flow in heterogeneous media. One is a Gaussian random field (GRF) with uncertain length scales, and the other is channelized permeability with the two regions defined by different GRFs. The obtained results indicate that the method allows high-dimensional parameter estimation while exhibiting stability, efficiency and accuracy.