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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2408.09821 |
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| _version_ | 1866917752107696128 |
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| author | Tapley, Benjamin K |
| author_facet | Tapley, Benjamin K |
| contents | We present and analyze a framework for designing symplectic neural networks (SympNets) based on geometric integrators for Hamiltonian differential equations. The SympNets are universal approximators in the space of Hamiltonian diffeomorphisms, interpretable and have a non-vanishing gradient property. We also give a representation theory for linear systems, meaning the proposed P-SympNets can exactly parameterize any symplectic map corresponding to quadratic Hamiltonians. Extensive numerical tests demonstrate increased expressiveness and accuracy -- often several orders of magnitude better -- for lower training cost over existing architectures. Lastly, we show how to perform symbolic Hamiltonian regression with SympNets for polynomial systems using backward error analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symplectic Neural Networks Based on Dynamical Systems Tapley, Benjamin K Machine Learning Computational Engineering, Finance, and Science Numerical Analysis Computational Physics We present and analyze a framework for designing symplectic neural networks (SympNets) based on geometric integrators for Hamiltonian differential equations. The SympNets are universal approximators in the space of Hamiltonian diffeomorphisms, interpretable and have a non-vanishing gradient property. We also give a representation theory for linear systems, meaning the proposed P-SympNets can exactly parameterize any symplectic map corresponding to quadratic Hamiltonians. Extensive numerical tests demonstrate increased expressiveness and accuracy -- often several orders of magnitude better -- for lower training cost over existing architectures. Lastly, we show how to perform symbolic Hamiltonian regression with SympNets for polynomial systems using backward error analysis. |
| title | Symplectic Neural Networks Based on Dynamical Systems |
| topic | Machine Learning Computational Engineering, Finance, and Science Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2408.09821 |