Lie-Poisson Neural Networks (LPNets): Data-Based Computing of
  Hamiltonian Systems with Symmetries
Christopher Eldred François Gay-Balmaz Sofiia Huraka Vakhtang Putkaradze
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.