Using MMD GANs to correct physics models and improve Bayesian parameter estimation

Bayesian parameter estimation methods are robust techniques for quantifying properties of physical systems which cannot be observed directly. In estimating such parameters, one first requires a physics model of the phenomenon to be studied. Often, such a model follows a series of assumptions to make parameter inference feasible. When simplified models are used for inference, however, systematic differences between model predictions and observed data may propagate throughout the parameter estimation process, biasing inference results. In this work, we use generative adversarial networks (GANs) based on the maximum mean discrepancy (MMD) to learn small stochastic corrections to physics models in order to minimize inference bias. We further propose a hybrid training procedure utilizing both the MMD and the standard GAN objective functionals. We demonstrate the ability to learn stochastic model corrections and eliminate inference bias on a toy problem wherein the true data distribution is known. Subsequently, we apply these methods to a mildly ill-posed inference problem in magnetic resonance imaging (MRI), showing improvement over an established inference method. Finally, because 3D MRI images often contain millions of voxels which would each require parameter inference, we train a conditional variational autoencoder (CVAE) network on the corrected MRI physics model to perform fast inference and make this approach practical.