Lie–Poisson Neural Networks (LPNets): Data-based computing of Hamiltonian systems with symmetries

Journal Article · Sat Feb 03 00:00:00 EST 2024 · Neural Networks

DOI:https://doi.org/10.1016/j.neunet.2024.106162· OSTI ID:2589933

Eldred, Christopher ^[1]; Gay-Balmaz, François ^[2]; Huraka, Sofiia ^[3]; Putkaradze, Vakhtang ^[3]

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1. Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
2. Nanyang Technological Univ. (Singapore)
3. Univ. of Alberta, Edmonton, AB (Canada)

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.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: NA0003525
• OSTI ID: 2589933
• Journal Information: Neural Networks, Journal Name: Neural Networks Vol. 173; ISSN 0893-6080
• Publisher: Elsevier BV
• Country of Publication: United States
• Language: English

Related Subjects

Data-based modeling
Hamiltonian systems
Long-term evolution
Neural equations
Poisson brackets