Gaussian processes for Bayesian inverse problems associated with linear partial differential equations

TL;DR

This paper explores the use of PDE-informed Gaussian priors as surrogate models for Bayesian inverse problems involving linear PDEs, demonstrating their effectiveness with limited training data through numerical experiments.

Contribution

It extends the framework of Raissi et al. (2017) to develop PDE-informed Gaussian priors for improved Bayesian inversion with scarce data.

Findings

PDE-informed Gaussian priors outperform traditional priors in Bayesian inverse problems.

Numerical experiments validate the superiority of the proposed approach.

The method is effective even with limited training data.

Abstract

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.