Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries
An accurate data-based prediction of the long-term evolution of Hamiltonian systems requires a network that preserves the appropriate structure under each time step. Every Hamiltonian system contains two essential ingredients: the Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetr...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2024
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| Online Access: | https://hdl.handle.net/10356/180076 |
| _version_ | 1826113809860788224 |
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| author | Eldred, Christopher Gay-Balmaz, François Huraka, Sofiia Putkaradze, Vakhtang |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Eldred, Christopher Gay-Balmaz, François Huraka, Sofiia Putkaradze, Vakhtang |
| author_sort | Eldred, Christopher |
| collection | NTU |
| description | An accurate data-based prediction of the long-term evolution of Hamiltonian systems requires a network that preserves the appropriate structure under each time step. Every Hamiltonian system contains two essential ingredients: the Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whose paradigm examples are the Lie-Poisson systems, have been shown to describe a broad category of physical phenomena, from satellite motion to underwater vehicles, fluids, geophysical applications, complex fluids, and plasma physics. The Poisson bracket in these systems comes from the symmetries, while the Hamiltonian comes from the underlying physics. We view the symmetry of the system as primary, hence the Lie-Poisson bracket is known exactly, whereas the Hamiltonian is regarded as coming from physics and is considered not known, or known approximately. Using this approach, we develop a network based on transformations that exactly preserve the Poisson bracket and the special functions of the Lie-Poisson systems (Casimirs) to machine precision. We present two flavors of such systems: one, where the parameters of transformations are computed from data using a dense neural network (LPNets), and another, where the composition of transformations is used as building blocks (G-LPNets). We also show how to adapt these methods to a larger class of Poisson brackets. We apply the resulting methods to several examples, such as rigid body (satellite) motion, underwater vehicles, a particle in a magnetic field, and others. The methods developed in this paper are important for the construction of accurate data-based methods for simulating the long-term dynamics of physical systems. |
| first_indexed | 2024-10-01T03:29:20Z |
| format | Journal Article |
| id | ntu-10356/180076 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:29:20Z |
| publishDate | 2024 |
| record_format | dspace |
| spelling | ntu-10356/1800762024-09-16T15:34:55Z Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries Eldred, Christopher Gay-Balmaz, François Huraka, Sofiia Putkaradze, Vakhtang School of Physical and Mathematical Sciences Mathematical Sciences Neural equations Data-based modeling An accurate data-based prediction of the long-term evolution of Hamiltonian systems requires a network that preserves the appropriate structure under each time step. Every Hamiltonian system contains two essential ingredients: the Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whose paradigm examples are the Lie-Poisson systems, have been shown to describe a broad category of physical phenomena, from satellite motion to underwater vehicles, fluids, geophysical applications, complex fluids, and plasma physics. The Poisson bracket in these systems comes from the symmetries, while the Hamiltonian comes from the underlying physics. We view the symmetry of the system as primary, hence the Lie-Poisson bracket is known exactly, whereas the Hamiltonian is regarded as coming from physics and is considered not known, or known approximately. Using this approach, we develop a network based on transformations that exactly preserve the Poisson bracket and the special functions of the Lie-Poisson systems (Casimirs) to machine precision. We present two flavors of such systems: one, where the parameters of transformations are computed from data using a dense neural network (LPNets), and another, where the composition of transformations is used as building blocks (G-LPNets). We also show how to adapt these methods to a larger class of Poisson brackets. We apply the resulting methods to several examples, such as rigid body (satellite) motion, underwater vehicles, a particle in a magnetic field, and others. The methods developed in this paper are important for the construction of accurate data-based methods for simulating the long-term dynamics of physical systems. Published version SH and VP were partially supported by the NSERC Discovery, Canada grant. 2024-09-16T00:43:24Z 2024-09-16T00:43:24Z 2024 Journal Article Eldred, C., Gay-Balmaz, F., Huraka, S. & Putkaradze, V. (2024). Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries. Neural Networks, 173, 106162-. https://dx.doi.org/10.1016/j.neunet.2024.106162 0893-6080 https://hdl.handle.net/10356/180076 10.1016/j.neunet.2024.106162 38335794 2-s2.0-85184770799 173 106162 en Neural Networks © 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). application/pdf |
| spellingShingle | Mathematical Sciences Neural equations Data-based modeling Eldred, Christopher Gay-Balmaz, François Huraka, Sofiia Putkaradze, Vakhtang Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries |
| title | Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries |
| title_full | Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries |
| title_fullStr | Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries |
| title_full_unstemmed | Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries |
| title_short | Lie-Poisson neural networks (LPNets): data-based computing of Hamiltonian systems with symmetries |
| title_sort | lie poisson neural networks lpnets data based computing of hamiltonian systems with symmetries |
| topic | Mathematical Sciences Neural equations Data-based modeling |
| url | https://hdl.handle.net/10356/180076 |
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