Bayesian PINNs for uncertainty-aware inverse problems (BPINN-IP)

TL;DR

This paper introduces BPINN-IP, a hierarchical Bayesian approach to Physics-Informed Neural Networks that incorporates prior knowledge and quantifies uncertainties in inverse problems, demonstrated through deconvolution and super-resolution tasks.

Contribution

It extends PINNs with a Bayesian framework for uncertainty quantification in linear inverse problems, using variational inference and Monte Carlo dropout.

Findings

Uncertainty quantification is effectively integrated into PINNs.

Preliminary results show promising application to deconvolution.

Method provides predictive means and variances for reconstructed images.

Abstract

The main contribution of this paper is to develop a hierarchical Bayesian formulation of PINNs for linear inverse problems, which is called BPINN-IP. The proposed methodology extends PINN to account for prior knowledge on the nature of the expected NN output, as well as its weights. Also, as we can have access to the posterior probability distributions, naturally uncertainties can be quantified. Also, variational inference and Monte Carlo dropout are employed to provide predictive means and variances for reconstructed images. Un example of applications to deconvolution and super-resolution is considered, details of the different steps of implementations are given, and some preliminary results are presented.