High precision PINNs in unbounded domains: application to singularity formulation in PDEs
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912446404362240 |
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| author | Wang, Yixuan Liu, Ziming Li, Zongyi Anandkumar, Anima Hou, Thomas Y. |
| author_facet | Wang, Yixuan Liu, Ziming Li, Zongyi Anandkumar, Anima Hou, Thomas Y. |
| contents | We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is $4$ digits smaller than that obtained in \cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High precision PINNs in unbounded domains: application to singularity formulation in PDEs Wang, Yixuan Liu, Ziming Li, Zongyi Anandkumar, Anima Hou, Thomas Y. Machine Learning Numerical Analysis We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is $4$ digits smaller than that obtained in \cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems. |
| title | High precision PINNs in unbounded domains: application to singularity formulation in PDEs |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2506.19243 |