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.

翻译：近期研究提供了令人信服的证据，表明物理信息神经网络（PINN）能高效求解偏微分方程（PDE）。然而，以往工作未能保证PINN在时空域上的最差残差误差——这一指标类似于数值求解器的容忍度，而仅关注其解与求解器在有限输入点上所得解之间的逐点比较。在实际应用中，不能认为在有限点集上的测试足以支撑部署，因为性能在不同点集上可能显著恶化。为解决此问题，我们建立了基于容忍度的PINN在整个输入域上的正确性条件。为验证这些条件的成立程度，我们提出$\partial$-CROWN：一种通用、高效且可扩展的训练后框架，用于约束PINN残差误差。通过将其应用于两个经典研究的PDE——伯格斯方程和薛定谔方程——以及两个更具挑战性、具有实际应用的方程——艾伦-卡恩方程和扩散-吸附方程——我们展示了该方法在获取严格证书方面的有效性。