On the contraction rate of the posterior distribution for nonlinear PDE parameter identification

Abstract: In this work, we investigate the estimation of a parameter $f$ in PDEs using Bayesian procedures, and focus on posterior distributions constructed using Gaussian process priors, and its variational approximation. We establish contraction rates for the posterior distribution and the variational approximation in the regime of low-regularity parameters. The main novelty of the study lies in relaxing the condition that the ground truth parameter must lie in the reproducing kernel Hilbert space of the Gaussian process prior, which is commonly imposed in existing studies on posterior contraction rate analysis [14,40,44]. The analysis relies on a delicate approximation argument that suitably balances various error sources. We illustrate the general theory on three nonlinear inverse problems for PDEs.