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

Area of Science:

• * Computational Physics
• * Applied Mathematics
• * Machine Learning

Background:

• * Hamiltonian systems are fundamental in describing physical phenomena, including satellite motion and fluid dynamics.
• * Accurate long-term predictions require computational methods that preserve the system's inherent mathematical structure.
• * Existing methods often struggle with long-term accuracy due to difficulties in preserving these structures.

Purpose of the Study:

• * To develop data-based predictive models for Hamiltonian systems that precisely preserve their underlying mathematical structure.
• * To create neural network architectures that respect the symmetries and invariants (Casimirs) of Lie-Poisson systems.
• * To enable highly accurate, long-term simulations of complex physical dynamics.

Main Methods:

• * Development of neural networks (LPNets) that learn transformations exactly preserving the Lie-Poisson bracket and Casimirs.
• * Introduction of G-LPNets, utilizing compositions of transformations as building blocks for enhanced structural preservation.
• * Adaptation of methods to handle a broader range of Poisson brackets.
• * Application to diverse physical systems like rigid body motion and particle dynamics in magnetic fields.

Main Results:

• * Achieved machine-precision preservation of Poisson brackets and Casimirs in learned transformations.
• * Demonstrated the efficacy of LPNets and G-LPNets in accurately simulating long-term dynamics.
• * Successfully applied the methods to various benchmark physical systems, validating their robustness.

Conclusions:

• * The proposed network-based approach offers a powerful tool for accurate, long-term simulations of Hamiltonian systems.
• * Preserving the fundamental mathematical structure, particularly symmetries, is crucial for reliable predictive modeling.
• * These methods advance the development of data-driven simulation techniques for complex physical applications.