发表机构
The University of Alabama（阿拉巴马大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出LLPNNs这一保结构框架，直接从可观测量学习李-泊松动力学，适用于正则与退化哈密顿系统，在三类系统中展现出优异预测精度与噪声鲁棒性。
AI 中文摘要
保结构神经网络是从数据中长期预测哈密顿系统的关键。力学与控制领域的诸多重要哈密顿系统可通过对称约化得到李-泊松系统，包括刚体、水下航行器、流体、等离子体及最优控制问题。学习此类系统的核心挑战在于其动力学在通常不可观测的动量变量上演化，而可用数据仅包含构型、速度等可观测量；在最优控制应用中，隐变量还包含不可观测的协态，且哈密顿量可能退化，导致对应拉格朗日量不存在，编码器-解码器方法无法适用。我们提出隐李-泊松神经网络（LLPNNs），这是一种直接从可观测量学习李-泊松动力学的保结构框架，该方法利用三类几何要素：（i）在主动变量上学习哈密顿解码器或伪拉格朗日编码器；（ii）通过李-泊松对称约化产生的通用诺特定理不变量构建隐轨迹；（iii）结合李-泊松流与基于Magnus的李群更新重构可观测量与隐动力学。该方法保留几何结构，适用于正则与退化哈密顿系统。我们在三个示例中验证：SO(3)上的广义刚体、SE(3)上的Kirchhoff水下航行器、SE(2)^N上的交互车辆最优控制问题。数值实验表明，该方法具备优异的长期预测精度、强噪声鲁棒性，且仅用适度数据集与轻量神经网络架构即可实现具竞争力的性能。
英文摘要
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.
Comments70 pages, 18 figures
