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CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetries

Christopher Eldred, François Gay-Balmaz, Vakhtang Putkaradze

TL;DR

CLPNets introduce a structure-preserving, data-efficient approach to learn the full phase-space dynamics of multi-part Hamiltonian systems with symmetries, including coupled $SO(3)$ and $SE(3)$ components. By constructing neural transforms from flows of test Hamiltonians that depend on momenta and group elements, the method preserves all Casimir invariants to machine precision and maintains Lie-Poisson structure, enabling reliable long-term predictions with relatively small datasets. The framework is demonstrated on two interacting rigid bodies and on coupled $SE(3)$ elements, achieving accurate dynamics with modest parameter counts (e.g., 108–189 parameters) and data (thousands of points), while exhibiting expected energy behavior and Lyapunov-consistent error growth in chaotic regimes. These results suggest a scalable path toward structure-preserving data-based simulations of complex elastic and robotic systems, with potential extensions to discretized continua and nonlinear elasticity problems.

Abstract

To accurately compute data-based prediction of Hamiltonian systems, especially the long-term evolution of such systems, it is essential to utilize methods that preserve the structure of the equations over time. We consider a case that is particularly challenging for data-based methods: systems with interacting parts that do not reduce to pure momentum evolution. Such systems are essential in scientific computations. For example, any discretization of a continuum elastic rod can be viewed as interacting elements that can move and rotate in space, with each discrete element moving on the group of rotations and translations $SE(3)$. We develop a novel method of data-based computation and complete phase space learning of such systems. We follow the original framework of \emph{SympNets} (Jin et al, 2020) building the neural network from canonical phase space mappings, and transformations that preserve the Lie-Poisson structure (\emph{LPNets}) as in (Eldred et al, 2024). We derive a novel system of mappings that are built into neural networks for coupled systems. We call such networks Coupled Lie-Poisson Neural Networks, or \emph{CLPNets}. We consider increasingly complex examples for the applications of CLPNets: rotation of two rigid bodies about a common axis, the free rotation of two rigid bodies, and finally the evolution of two connected and interacting $SE(3)$ components. Our method preserves all Casimir invariants of each system to machine precision, irrespective of the quality of the training data, and preserves energy to high accuracy. Our method also shows good resistance to the curse of dimensionality, requiring only a few thousand data points for all cases studied, with the effective dimension varying from three to eighteen. Additionally, the method is highly economical in memory requirements, requiring only about 200 parameters for the most complex case considered.

CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetries

TL;DR

CLPNets introduce a structure-preserving, data-efficient approach to learn the full phase-space dynamics of multi-part Hamiltonian systems with symmetries, including coupled and components. By constructing neural transforms from flows of test Hamiltonians that depend on momenta and group elements, the method preserves all Casimir invariants to machine precision and maintains Lie-Poisson structure, enabling reliable long-term predictions with relatively small datasets. The framework is demonstrated on two interacting rigid bodies and on coupled elements, achieving accurate dynamics with modest parameter counts (e.g., 108–189 parameters) and data (thousands of points), while exhibiting expected energy behavior and Lyapunov-consistent error growth in chaotic regimes. These results suggest a scalable path toward structure-preserving data-based simulations of complex elastic and robotic systems, with potential extensions to discretized continua and nonlinear elasticity problems.

Abstract

To accurately compute data-based prediction of Hamiltonian systems, especially the long-term evolution of such systems, it is essential to utilize methods that preserve the structure of the equations over time. We consider a case that is particularly challenging for data-based methods: systems with interacting parts that do not reduce to pure momentum evolution. Such systems are essential in scientific computations. For example, any discretization of a continuum elastic rod can be viewed as interacting elements that can move and rotate in space, with each discrete element moving on the group of rotations and translations . We develop a novel method of data-based computation and complete phase space learning of such systems. We follow the original framework of \emph{SympNets} (Jin et al, 2020) building the neural network from canonical phase space mappings, and transformations that preserve the Lie-Poisson structure (\emph{LPNets}) as in (Eldred et al, 2024). We derive a novel system of mappings that are built into neural networks for coupled systems. We call such networks Coupled Lie-Poisson Neural Networks, or \emph{CLPNets}. We consider increasingly complex examples for the applications of CLPNets: rotation of two rigid bodies about a common axis, the free rotation of two rigid bodies, and finally the evolution of two connected and interacting components. Our method preserves all Casimir invariants of each system to machine precision, irrespective of the quality of the training data, and preserves energy to high accuracy. Our method also shows good resistance to the curse of dimensionality, requiring only a few thousand data points for all cases studied, with the effective dimension varying from three to eighteen. Additionally, the method is highly economical in memory requirements, requiring only about 200 parameters for the most complex case considered.
Paper Structure (42 sections, 1 theorem, 80 equations, 9 figures)

This paper contains 42 sections, 1 theorem, 80 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Physical relevance and importance for applications
  3. Relevant work related to data-based computing of Poisson systems
  4. Canonical and non-canonical Hamiltonian systems
  5. Canonical Hamiltonian Systems
  6. General Poisson brackets
  7. First approach: discovering the Hamiltonian system
  8. Second approach: Learning the structure-preserving mappings of phase space for canonical systems
  9. Learning structure-preserving mappings for non-canonical Hamiltonian systems
  10. Statement of the problem
  11. Novelty of this work
  12. General setting for interacting systems on Lie groups
  13. Equations of motion
  14. Hamiltonian formulation and Poisson brackets
  15. Defining the building blocks of Neural Network using transformations generated by test Hamiltonians
  16. ...and 27 more sections

Key Result

Lemma A.1

Let $C_ \mathfrak{g} : \mathfrak{g} ^* \rightarrow \mathbb{R}$ be a Casimir function of the Lie-Poisson bracket on $\mathfrak{g} ^*$, i.e., Then the function $C: \mathfrak{g} ^* \times \mathfrak{g} ^* \times G \rightarrow \mathbb{R}$ defined by is a Casimir function for the Poisson bracket reduced_Poisson.

Figures (9)

  • Figure 1: Two rigid bodies $\mathcal{B}_{1,2}$ rotating about a common point which is fixed in space. Each rigid body $i=1,2$ contains several charges $q_{i,j}$, $j=1, \ldots,m$, which are located in the coordinates $\boldsymbol{\xi}_{i,j}$ in the body frame. Two charges on each body are shown on this illustration.
  • Figure 2: The results of simulations of equations \ref{['SO3_coupled_equations']} in the reduced case of $SO(2)$ rotations about the vertical axis. Blue lines: ground truth; red lines: predictions provided by CLPNets.
  • Figure 3: The comparison of the Casimirs (left), energy (center), as well as the computation of the Mean Absolute Error (right) for all components for the simulations presented on Figure \ref{['fig:momenta_orientation_SO2']}. The long-term energy conservation for the neural network, although oscillatory, conserves the energy on average with high precision.
  • Figure 4: The results of simulations of equations \ref{['SO3_coupled_equations']} in the general case of $SO(3)$ rotations. As before, blue lines are the ground truth; red lines are predictions by CLPNets. All components of angular momenta $\boldsymbol{\mu}_{1,2}$ and three components of orientation matrix $p_{1k}$, $k=1,2,3$ are shown.
  • Figure 5: The comparison of the Casimirs (left), energy (center), and the Mean Absolute Error for all components of momenta and three components of $p_{ij}$ for the simulations presented on Figure \ref{['fig:momenta_orientation_SO3']}. The energy and Casimir are shown over all the time interval computed (5000 data points, or $t=500$). The MAE on the right panel is computed only for the time interval corresponding to $t=50$ (500 data points). The growth of errors corresponding to the Lyapunov exponent on the right panel is demonstrated with a dashed red line.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Remark 4.1: On the canonical structure of the equations
  • Remark 5.1: On resisting the curse of dimensionality by CLPNets
  • Lemma A.1