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.

翻译：精确基于数据的哈密顿系统长期演化预测，需要网络在每一步时间迭代中保持适当结构。每个哈密顿系统包含两个核心要素：泊松括号与哈密顿量。具有对称性的哈密顿系统（其典型范例为李-泊松系统）已被证明能描述从卫星运动、水下航行器到流体、地球物理应用、复杂流体乃至等离子体物理等广泛物理现象。此类系统中的泊松括号源于对称性，而哈密顿量则来自底层物理规律。我们将系统对称性视为首要因素，因此精确已知李-泊松括号，而将哈密顿量视为源自物理规律且未知（或近似已知）。基于此思路，我们构建了基于保结构变换的网络，能够精确保持泊松括号与李-泊松系统特殊函数（卡西米尔不变量）至机器精度。我们提出两种实现形式：其一通过稠密神经网络从数据中计算变换参数（LPNets），其二以变换复合作为基本构建模块（G-LPNets）。同时展示了如何将这些方法推广至更广泛的泊松括号类型。我们将所提方法应用于刚体（卫星）运动、水下航行器、磁场中粒子运动等多个实例。本文发展的方法对构建物理系统长期动力学精确数据驱动模拟方法具有重要意义。