Efficient Error Certification for Physics-Informed Neural Networks
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914815116574720 |
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| author | Eiras, Francisco Bibi, Adel Bunel, Rudy Dvijotham, Krishnamurthy Dj Torr, Philip Kumar, M. Pawan |
| author_facet | Eiras, Francisco Bibi, Adel Bunel, Rudy Dvijotham, Krishnamurthy Dj Torr, Philip Kumar, M. Pawan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_10157 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficient Error Certification for Physics-Informed Neural Networks Eiras, Francisco Bibi, Adel Bunel, Rudy Dvijotham, Krishnamurthy Dj Torr, Philip Kumar, M. Pawan Machine Learning Mathematical Physics 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. |
| title | Efficient Error Certification for Physics-Informed Neural Networks |
| topic | Machine Learning Mathematical Physics |
| url | https://arxiv.org/abs/2305.10157 |