  This work presents a general geometric framework for simulating and learning
the dynamics of Hamiltonian systems that are invariant under a Lie group of
transformations. This means that a group of symmetries is known to act on the
system respecting its dynamics and, as a consequence, Noether's Theorem,
conserved quantities are observed. We propose to simulate and learn the
mappings of interest through the construction of $G$-invariant Lagrangian
submanifolds, which are pivotal objects in symplectic geometry. A notable
property of our constructions is that the simulated/learned dynamics also
preserves the same conserved quantities as the original system, resulting in a
more faithful surrogate of the original dynamics than non-symmetry aware
methods, and in a more accurate predictor of non-observed trajectories.
Furthermore, our setting is able to simulate/learn not only Hamiltonian flows,
but any Lie group-equivariant symplectic transformation. Our designs leverage
pivotal techniques and concepts in symplectic geometry and geometric mechanics:
reduction theory, Noether's Theorem, Lagrangian submanifolds, momentum
mappings, and coisotropic reduction among others. We also present methods to
learn Poisson transformations while preserving the underlying geometry and how
to endow non-geometric integrators with geometric properties. Thus, this work
presents a novel attempt to harness the power of symplectic and Poisson
geometry towards simulating and learning problems.
