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

arXiv: https://arxiv.org/abs/2408.16160 · HTML: https://mlsanity.com/paper/2408.16160/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 50 references that are in the map.
- First posted on arXiv 28 Aug 2024.

## Builds on (most cited first, 11 of 11)

- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Symmetry Preservation in Hamiltonian Systems: Simulation and Learning](https://mlsanity.com/paper/2308.16331/) (2023)
- [Designing Poisson Integrators Through Machine Learning](https://mlsanity.com/paper/2403.20139/) (2024)

## Built on by (most cited first, 3 of 3)

- [Learning Nonholonomic Dynamics with Constraint Discovery](https://mlsanity.com/paper/2410.15201/) (2024)
- [Variational Neural Networks for Observable Thermodynamics (V-NOTS)](https://mlsanity.com/paper/2509.09899/) (2025)
- [Symplectic Representation of Legendre Dynamics](https://mlsanity.com/paper/2512.19409/) (2025)

## Abstract (arXiv)

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 SympNets (Jin et al, 2020) building the neural network from canonical phase space mappings, and transformations that preserve the Lie-Poisson structure (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 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.