Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs

TL;DR

This paper establishes statistical guarantees for Bayesian PINNs solving elliptic PDEs, showing the posterior concentrates near the true solution at near-minimax rates with adaptive priors.

Contribution

It proves the posterior contraction rates for Bayesian PINNs applied to elliptic PDEs, including rate-adaptivity and uncertainty quantification.

Findings

Posterior concentrates at near-minimax rates.

Prior is rate-adaptive without smoothness knowledge.

Provides statistical guarantees for uncertainty quantification.

Abstract

We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous Dirichlet boundary condition from independent and noisy measurements collected both inside the domain and on the boundary. Assuming that the PDE admits a strong solution in a H\"older space and using with a suitably constructed prior on the neural network weights, we prove that the posterior distribution concentrates around the exact solution at a near-minimax rate. Furthermore, the chosen prior is rate-adaptive: the posterior contracts at an (almost) optimal rate without prior knowledge of the smoothness level of the exact solution. Our results provide statistical guarantees for uncertainty quantification of PDEs via Bayesian PINNs.