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.

本研究解决了数据驱动的哈密顿系统长期演化预测中保持方程结构的问题，尤其是处理多个相互作用部分的系统。提出的耦合李-波松神经网络（CLPNets）通过保留李-波松结构的映射，有效学习并计算相空间，显著提高了系统能量的准确性和稳定性，同时具有较低的训练数据需求和记忆消耗。

CLPNets：针对具有对称性的多部分哈密顿系统的耦合李-波松神经网络