Gaussian processes for Bayesian inverse problems associated with linear partial differential equations
This work addresses inverse problems in computational science, offering an incremental improvement for scenarios with sparse data.
The authors tackled the challenge of Bayesian inverse problems for linear PDEs with limited training data by developing PDE-informed Gaussian priors, demonstrating their superiority over traditional priors in numerical experiments.
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.