Efficient Error Certification for Physics-Informed Neural Networks

Recent work provides promising evidence that Physics-Informed Neural Networks (PINN) can efficiently solve 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 on a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment, as the performance could be substantially worse on a different set. To alleviate this issue, we establish guaranteed error-based conditions for PINNs over their continuous applicability domain. To verify the extent to which they hold, we introduce $\partial$-CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classically studied PINNs - Burgers' and Schr\"odinger's equations -, and two more challenging ones with real-world applications - the Allan-Cahn and Diffusion-Sorption equations.