# Model constrained empirical Bayesian neural networks for inverse problems

## Palavras-chave:

Deep learning, inverse problems, Bayesian methods, uncertainty quantification, variational inference## Resumo

Principled Uncertainty quantification (UQ) in deep learning is still an unsolved problem. Numerous methods have been developed so far, with Bayesian neural networks (BNNs) as the popular approach. BNNs, while inherently UQ-enabled and resistant to over-fitting, suffer from unnatural and artificial priors over their parameters. This paper develops a model-constrained framework for quantifying the uncertainty in deep neural network inverse solutions. At the heart of our approach is an interpretable and physically-meaningful prior over neural network parameters trained through use of Stein variational gradient descent (SVGD). We provide comprehensive numerical results for a 2D inverse heat conductivity problem and a 2D inverse initial conditions problems for both the time-dependent Burgers’ and Navier-Stokes equations.