Provably Correct Physics-Informed Neural Networks

Physics-informed neural networks (PINN) have been proven efficient at solving partial differential equations (PDE). However, previous works have failed to provide guarantees on the worst-case residual error of a PINN across the spatio-temporal domain – a measure akin to the tolerance of numerical solvers – focusing instead on point-wise comparisons between their solution and the ones obtained by a solver at a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment. To alleviate this issue, we establish tolerance-based correctness conditions for PINNs over the entire input domain. To verify the extent to which they hold, we introduce ∂-CROWN: a general and efficient post-training framework to bound PINN errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classical PINNs – Burgers’ and Schrödinger's equations –, and two more challenging ones – the Allan-Cahn and Diffusion-Sorption equations.