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 tolerance-based correctness conditions for PINNs over the entire input 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 PDEs - Burgers' and Schr\"odinger's equations -, and two more challenging ones with real-world applications - the Allan-Cahn and Diffusion-Sorption equations.

本文介绍了一种基于物理信息的神经网络（PINN）来解决偏微分方程的方法，并提出了一种基于容差的正确性条件的后训练框架（CROWN），用于限制PINN残差误差，并在经典PDE和现实应用中进行了实际效果测试。

可证明正确的物理信息神经网络