Efficient Error Certification for Physics-Informed Neural Networks

Francisco Eiras, Adel Bibi, Rudy R Bunel, Krishnamurthy Dj Dvijotham, Philip Torr, M. Pawan Kumar
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:12318-12347, 2024.

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 Schrödinger’s equations –, and two more challenging ones with real-world applications – the Allan-Cahn and Diffusion-Sorption equations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-eiras24a, title = {Efficient Error Certification for Physics-Informed Neural Networks}, author = {Eiras, Francisco and Bibi, Adel and Bunel, Rudy R and Dvijotham, Krishnamurthy Dj and Torr, Philip and Kumar, M. Pawan}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {12318--12347}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/eiras24a/eiras24a.pdf}, url = {https://proceedings.mlr.press/v235/eiras24a.html}, 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 Schrödinger’s equations –, and two more challenging ones with real-world applications – the Allan-Cahn and Diffusion-Sorption equations.} }
Endnote
%0 Conference Paper %T Efficient Error Certification for Physics-Informed Neural Networks %A Francisco Eiras %A Adel Bibi %A Rudy R Bunel %A Krishnamurthy Dj Dvijotham %A Philip Torr %A M. Pawan Kumar %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-eiras24a %I PMLR %P 12318--12347 %U https://proceedings.mlr.press/v235/eiras24a.html %V 235 %X 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 Schrödinger’s equations –, and two more challenging ones with real-world applications – the Allan-Cahn and Diffusion-Sorption equations.
APA
Eiras, F., Bibi, A., Bunel, R.R., Dvijotham, K.D., Torr, P. & Kumar, M.P.. (2024). Efficient Error Certification for Physics-Informed Neural Networks. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:12318-12347 Available from https://proceedings.mlr.press/v235/eiras24a.html.

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