On the estimation rate of Bayesian PINN for inverse problems

Abstract: Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for PINNs that work remarkably in practice, our theoretical understanding of their performances is somewhat limited. In this work, we study the behavior of a Bayesian PINN estimator of the solution of a PDE from $n$ independent noisy measurement of the solution. We focus on a class of equations that are linear in their parameters (with unknown coefficients $\theta_\star$). We show that when the partial differential equation admits a classical solution (say $u_\star$), differentiable to order $\beta$, the mean square error of the Bayesian posterior mean is at least of order $n^{-2\beta/(2\beta + d)}$. Furthermore, we establish a convergence rate of the linear coefficients of $\theta_\star$ depending on the order of the underlying differential operator. Last but not least, our theoretical results are validated through extensive simulations.