Latent Lie-Poisson Neural Networks Learn Complex System Dynamics

This research introduces Latent Lie-Poisson Neural Networks (LLPNNs), a novel framework for learning the dynamics of complex Hamiltonian systems directly from observable data. Many critical systems in mechanics and control, such as rigid bodies or underwater vehicles, can be described by Lie-Poisson systems, but their underlying momentum variables are often unobservable. Traditional methods struggle with this challenge, especially when the Hamiltonian is degenerate. LLPNNs overcome this by integrating three geometric principles: learning a Hamiltonian decoder or pseudo-Lagrangian encoder, constructing latent trajectories via a universal Noether invariant, and reconstructing dynamics through Lie-Poisson flows combined with Magnus-based Lie-group updates. The method preserves the geometric structure of the system, making it applicable to both regular and degenerate Hamiltonian systems. Numerical experiments on examples like a generalized rigid body and an optimal control problem show excellent long-term predictive accuracy, strong noise robustness, and competitive performance with minimal data and lightweight neural networks.