Gaussian processes for Bayesian inverse problems associated with linear
  partial differential equations
Tianming Bai Aretha L. Teckentrup Konstantinos C. Zygalakis
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.