On the consistency of the posterior distribution for nonlinear PDE parameter identification

In this work, we investigate the estimation of a parameter 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. Specifically, the ground truth is only assumed to belong to the support space of the prior rather than to its associated reproducing kernel Hilbert space. Also we derive contraction rates for the posterior distribution constructed using the randomly truncated Gaussian process prior. The analysis relies on a delicate approximation argument that approximates low-regularity ground truths by suitable elements in the reproducing kernel Hilbert space and balances various error sources. We illustrate the general theory on three nonlinear inverse problems for PDEs.