Error estimates of physics-informed neural networks for approximating Boltzmann equation
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866914163646791680 |
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| author | Abdo, Elie Chai, Lihui Hu, Ruimeng Yang, Xu |
| author_facet | Abdo, Elie Chai, Lihui Hu, Ruimeng Yang, Xu |
| contents | Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57], we provide a rigorous error analysis for PINNs in approximating the solution of the Boltzmann equation near a global Maxwellian. The challenge arises from the nonlocal quadratic interaction term defined in the unbounded domain of velocity space. Analyzing this term on an unbounded domain requires the inclusion of a truncation function, which demands delicate analysis techniques. As a generalization of this analysis, we also provide proof of the asymptotic preserving property when using micro-macro decomposition-based neural networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Error estimates of physics-informed neural networks for approximating Boltzmann equation Abdo, Elie Chai, Lihui Hu, Ruimeng Yang, Xu Numerical Analysis Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57], we provide a rigorous error analysis for PINNs in approximating the solution of the Boltzmann equation near a global Maxwellian. The challenge arises from the nonlocal quadratic interaction term defined in the unbounded domain of velocity space. Analyzing this term on an unbounded domain requires the inclusion of a truncation function, which demands delicate analysis techniques. As a generalization of this analysis, we also provide proof of the asymptotic preserving property when using micro-macro decomposition-based neural networks. |
| title | Error estimates of physics-informed neural networks for approximating Boltzmann equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.08383 |