Enhanced Uncertainty Quantification Variational Autoencoders for the Solution of Bayesian Inverse Problems

Abstract

Among other uses, neural networks are used to solve deterministic and Bayesian inverse problems, where variational autoencoders enable the Bayesian estimation of model parameters and their distribution from observational data allowing real-time inverse uncertainty quantification. In this work, we build upon existing research by proposing a novel loss function to train variational autoencoders. For affine forward maps, we theoretically prove that an affine transformation of the latent states of the trained variational autoencoder represents the mean and covariance of the posterior distribution of the model parameters. We validate this result through numerical tests and we show the superior accuracy and generalization performance of the proposed variational autoencoder compared with the original formulation. We test the proposed variational autoencoder on a Laplace equation, comparing it to the original formulation and to Markov chain Monte Carlo methods. The proposed variational autoencoder outperforms the original formulation in terms of accuracy and yields slightly lower errors than Markov chain Monte Carlo in estimating the posterior variance over 100 in silico generated samples, although it exhibits slightly higher errors in the posterior mean estimate. Moreover, the variational autoencoder enables the solution of Bayesian inverse problems nearly 800 times faster than Markov chain Monte Carlo.