An accurate data-based prediction of the long-term evolution of Hamiltonian
systems requires a network that preserves the appropriate structure under each
time step. Every Hamiltonian system contains two essential ingredients: the
Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whose
paradigm examples are the Lie-Poisson systems, have been shown to describe a
broad category of physical phenomena, from satellite motion to underwater
vehicles, fluids, geophysical applications, complex fluids, and plasma physics.
The Poisson bracket in these systems comes from the symmetries, while the
Hamiltonian comes from the underlying physics. We view the symmetry of the
system as primary, hence the Lie-Poisson bracket is known exactly, whereas the
Hamiltonian is regarded as coming from physics and is considered not known, or
known approximately. Using this approach, we develop a network based on
transformations that exactly preserve the Poisson bracket and the special
functions of the Lie-Poisson systems (Casimirs) to machine precision. We
present two flavors of such systems: one, where the parameters of
transformations are computed from data using a dense neural network (LPNets),
and another, where the composition of transformations is used as building
blocks (G-LPNets). We also show how to adapt these methods to a larger class of
Poisson brackets. We apply the resulting methods to several examples, such as
rigid body (satellite) motion, underwater vehicles, a particle in a magnetic
field, and others. The methods developed in this paper are important for the
construction of accurate data-based methods for simulating the long-term
dynamics of physical systems.

基于数据的哈密顿系统长期演化精确预测需要一个在每个时间步骤下保留适当结构的网络。本文介绍了两种确切保留李 - 泊松系统的 Poisson 括号和特殊函数（卡西米尔）到机器精度的转换，一种是使用密集神经网络计算转换参数的 LPNets，另一种是使用转换的组合作为构建模块的 G-LPNets，并展示了如何将这些方法应用于更大类别的 Poisson 括号。这些方法对于构建准确的基于数据的物理系统长期动力学仿真方法至关重要。