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

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.

Many scientific problems — from heat flow to groundwater movement — are described by equations called partial differential equations (PDEs). A popular method called a Physics-Informed Neural Network (PINN) trains a network to fit data while obeying the underlying physics, but typically gives only a single "best guess" with no measure of reliability. The Bayesian version of PINNs instead returns a range of plausible solutions, naturally quantifying uncertainty. Though useful in practice, these methods have lacked rigorous theoretical support. Our paper provides one of the first such guarantees. For a broad class of PDEs, we prove that as more noisy measurements accumulate, the Bayesian PINN's plausible solutions reliably concentrate around the true solution at essentially the fastest possible rate — and this happens automatically, without needing to know in advance how smooth the solution is. Our results help justify the use of Bayesian PINNs in scientific applications where trustworthy uncertainty estimates matter.