  We introduce Sparse Symplectically Integrated Neural Networks (SSINNs), a
novel model for learning Hamiltonian dynamical systems from data. SSINNs
combine fourth-order symplectic integration with a learned parameterization of
the Hamiltonian obtained using sparse regression through a mathematically
elegant function space. This allows for interpretable models that incorporate
symplectic inductive biases and have low memory requirements. We evaluate
SSINNs on four classical Hamiltonian dynamical problems: the H\'enon-Heiles
system, nonlinearly coupled oscillators, a multi-particle mass-spring system,
and a pendulum system. Our results demonstrate promise in both system
prediction and conservation of energy, often outperforming the current
state-of-the-art black-box prediction techniques by an order of magnitude.
Further, SSINNs successfully converge to true governing equations from highly
limited and noisy data, demonstrating potential applicability in the discovery
of new physical governing equations.
