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An accurate data-based prediction of the long-term evolution of Hamiltoniansystems requires a network that preserves the appropriate structure under eachtime step. Every Hamiltonian system contains two essential ingredients: thePoisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whoseparadigm examples are the Lie-Poisson systems, have been shown to describe abroad category of physical phenomena, from satellite motion to underwatervehicles, fluids, geophysical applications, complex fluids, and plasma physics.The Poisson bracket in these systems comes from the symmetries, while theHamiltonian comes from the underlying physics. We view the symmetry of thesystem as primary, hence the Lie-Poisson bracket is known exactly, whereas theHamiltonian is regarded as coming from physics and is considered not known, orknown approximately. Using this approach, we develop a network based ontransformations that exactly preserve the Poisson bracket and the specialfunctions of the Lie-Poisson systems (Casimirs) to machine precision. Wepresent two flavors of such systems: one, where the parameters oftransformations are computed from data using a dense neural network (LPNets),and another, where the composition of transformations is used as buildingblocks (G-LPNets). We also show how to adapt these methods to a larger class ofPoisson brackets. We apply the resulting methods to several examples, such asrigid body (satellite) motion, underwater vehicles, a particle in a magneticfield, and others. The methods developed in this paper are important for theconstruction of accurate data-based methods for simulating the long-termdynamics of physical systems.

Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries