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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.09048 |
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| _version_ | 1866913158587744256 |
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| author | Baez, Anthony Zhang, Wang Ma, Ziwen Nguyen, Lam Das, Subhro Daniel, Luca |
| author_facet | Baez, Anthony Zhang, Wang Ma, Ziwen Nguyen, Lam Das, Subhro Daniel, Luca |
| contents | We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks Baez, Anthony Zhang, Wang Ma, Ziwen Nguyen, Lam Das, Subhro Daniel, Luca Machine Learning We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists. |
| title | Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.09048 |