Efficient Error Certification for Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2305.10157v2
• Date: Wed, 29 May 2024 11:08:06 GMT
• Title: Efficient Error Certification for Physics-Informed Neural Networks
• Authors: Francisco Eiras, Adel Bibi, Rudy Bunel, Krishnamurthy Dj Dvijotham, Philip Torr, M. Pawan Kumar,
• Abstract summary: We introduce $partial$-CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining certificates tight by applying it to two classically studied PINNs and two more challenging ones with real-world applications.
• Score: 25.712851771991218
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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