Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics

2026-07-30

AI Summary

This work addresses the challenge of learning dynamics in Lie-Poisson systems where only configuration and velocity are observable, momentum is latent, and the Hamiltonian may be degenerate—settings where conventional structure-preserving methods fail. The authors propose a geometric structure-preserving neural framework that constructs latent variables using Noether invariants derived from Lie-Poisson symmetry reduction, combined with either a Hamiltonian decoder or a pseudo-Lagrangian encoder. Observable and latent dynamics are jointly reconstructed by integrating Lie-Poisson flows with Magnus-type Lie group integrators. Notably, this approach enables direct data-driven learning of latent dynamics for both degenerate and non-degenerate Lie-Poisson systems without requiring an explicit Lagrangian. It demonstrates superior long-term prediction accuracy, robustness to strong noise, and sample efficiency in benchmark tasks including SO(3) rigid body dynamics, SE(3) underwater vehicles, and SE(2)^N multi-agent optimal control.

Abstract

Structure-preserving neural networks are essential for the long-term prediction of Hamiltonian systems from data. Many important Hamiltonian systems in mechanics and control admit symmetry reduction to Lie--Poisson systems, including rigid bodies, underwater vehicles, fluids, plasmas, and optimal control problems. A fundamental challenge in learning such systems is that their dynamics evolve in momentum variables that are typically unobservable, while available data consist only of observable quantities such as configurations and velocities. In optimal control applications, the situation is further complicated because the latent variables contain unobservable co-states and the Hamiltonian may be degenerate, preventing the existence of a corresponding Lagrangian and rendering the encoder-decoder approaches inapplicable. We introduce Latent Lie--Poisson Neural Networks (LLPNNs), a structure-preserving framework for learning Lie--Poisson dynamics directly from observable data. The proposed approach exploits three geometric ingredients: (i) learning either a Hamiltonian decoder or a pseudo-Lagrangian encoder on the active variables, (ii) constructing latent trajectories through a universal Noether invariant arising from Lie--Poisson symmetry reduction, and (iii) reconstructing observable and latent dynamics through Lie--Poisson flows combined with Magnus-based Lie-group updates. The resulting method preserves the geometric structure and is applicable to both regular and degenerate Hamiltonian systems. We demonstrate the method on three examples: a generalized rigid body on SO(3), Kirchhoff's underwater vehicle on SE(3), and an optimal-control problem for interacting vehicles on $SE(2)^N$. Numerical experiments show excellent long-term predictive accuracy, strong robustness to noise, and competitive performance using only modest datasets and lightweight neural-network architectures.

Problem

• Lie-Poisson systems
• latent dynamics
• observable data
• degenerate Hamiltonian
• symmetry reduction

Innovation

• Latent Lie-Poisson Neural Networks
• structure-preserving learning
• symmetry reduction
• Noether invariant
• Magnus integrator