  We propose a novel modular inference approach combining two different
generative models -- generative adversarial networks (GAN) and normalizing
flows -- to approximate the posterior distribution of physics-based Bayesian
inverse problems framed in high-dimensional ambient spaces. We dub the proposed
framework GAN-Flow. The proposed method leverages the intrinsic dimension
reduction and superior sample generation capabilities of GANs to define a
low-dimensional data-driven prior distribution. Once a trained GAN-prior is
available, the inverse problem is solved entirely in the latent space of the
GAN using variational Bayesian inference with normalizing flow-based
variational distribution, which approximates low-dimensional posterior
distribution by transforming realizations from the low-dimensional latent prior
(Gaussian) to corresponding realizations of a low-dimensional variational
posterior distribution. The trained GAN generator then maps realizations from
this approximate posterior distribution in the latent space back to the
high-dimensional ambient space. We also propose a two-stage training strategy
for GAN-Flow wherein we train the two generative models sequentially.
Thereafter, GAN-Flow can estimate the statistics of posterior-predictive
quantities of interest at virtually no additional computational cost. The
synergy between the two types of generative models allows us to overcome many
challenges associated with the application of Bayesian inference to large-scale
inverse problems, chief among which are describing an informative prior and
sampling from the high-dimensional posterior. We demonstrate the efficacy and
flexibility of GAN-Flow on various physics-based inverse problems of varying
ambient dimensionality and prior knowledge using different types of GANs and
normalizing flows.
