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

2026-05-09

AI Summary

This work addresses the solution and uncertainty quantification of nonhomogeneous Dirichlet elliptic partial differential equations from noisy interior and boundary observations. It proposes a Bayesian physics-informed neural network (PINN) approach that infers strong PDE solutions from stochastically perturbed data by endowing neural network weights with tailored priors. The study establishes, for the first time, a posterior contraction theory for Bayesian PINNs, rigorously proving that the posterior distribution concentrates around the true solution at a rate nearly attaining the minimax optimal bound. Notably, this convergence is adaptive—it does not require prior knowledge of the solution’s smoothness. The framework thus provides a statistically sound foundation for PINN-based PDE solvers, offering both rigorous theoretical guarantees and reliable 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 Hölder 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.

Problem

• Bayesian PINNs
• elliptic PDEs
• posterior concentration
• uncertainty quantification
• noisy measurements

Innovation

• Bayesian PINNs
• posterior contraction
• elliptic PDEs
• rate-adaptive prior
• uncertainty quantification