Preconditioning for Physics-Informed Neural Networks
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arXiv
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| Autori principali: | , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909090109718528 |
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| author | Liu, Songming Su, Chang Yao, Jiachen Hao, Zhongkai Su, Hang Wu, Youjia Zhu, Jun |
| author_facet | Liu, Songming Su, Chang Yao, Jiachen Hao, Zhongkai Su, Hang Wu, Youjia Zhu, Jun |
| contents | Physics-informed neural networks (PINNs) have shown promise in solving various partial differential equations (PDEs). However, training pathologies have negatively affected the convergence and prediction accuracy of PINNs, which further limits their practical applications. In this paper, we propose to use condition number as a metric to diagnose and mitigate the pathologies in PINNs. Inspired by classical numerical analysis, where the condition number measures sensitivity and stability, we highlight its pivotal role in the training dynamics of PINNs. We prove theorems to reveal how condition number is related to both the error control and convergence of PINNs. Subsequently, we present an algorithm that leverages preconditioning to improve the condition number. Evaluations of 18 PDE problems showcase the superior performance of our method. Significantly, in 7 of these problems, our method reduces errors by an order of magnitude. These empirical findings verify the critical role of the condition number in PINNs' training. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00531 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Preconditioning for Physics-Informed Neural Networks Liu, Songming Su, Chang Yao, Jiachen Hao, Zhongkai Su, Hang Wu, Youjia Zhu, Jun Machine Learning Numerical Analysis Physics-informed neural networks (PINNs) have shown promise in solving various partial differential equations (PDEs). However, training pathologies have negatively affected the convergence and prediction accuracy of PINNs, which further limits their practical applications. In this paper, we propose to use condition number as a metric to diagnose and mitigate the pathologies in PINNs. Inspired by classical numerical analysis, where the condition number measures sensitivity and stability, we highlight its pivotal role in the training dynamics of PINNs. We prove theorems to reveal how condition number is related to both the error control and convergence of PINNs. Subsequently, we present an algorithm that leverages preconditioning to improve the condition number. Evaluations of 18 PDE problems showcase the superior performance of our method. Significantly, in 7 of these problems, our method reduces errors by an order of magnitude. These empirical findings verify the critical role of the condition number in PINNs' training. |
| title | Preconditioning for Physics-Informed Neural Networks |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2402.00531 |