SetPINNs: Set-based Physics-informed Neural Networks
Fuente:
arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866918031193538560 |
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| author | Nagda, Mayank Ostheimer, Phil Specht, Thomas Rhein, Frank Jirasek, Fabian Mandt, Stephan Kloft, Marius Fellenz, Sophie |
| author_facet | Nagda, Mayank Ostheimer, Phil Specht, Thomas Rhein, Frank Jirasek, Fabian Mandt, Stephan Kloft, Marius Fellenz, Sophie |
| contents | Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SetPINNs: Set-based Physics-informed Neural Networks Nagda, Mayank Ostheimer, Phil Specht, Thomas Rhein, Frank Jirasek, Fabian Mandt, Stephan Kloft, Marius Fellenz, Sophie Machine Learning Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness. |
| title | SetPINNs: Set-based Physics-informed Neural Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2409.20206 |