  Predicting the behaviors of Hamiltonian systems has been drawing increasing
attention in scientific machine learning. However, the vast majority of the
literature was focused on predicting separable Hamiltonian systems with their
kinematic and potential energy terms being explicitly decoupled while building
data-driven paradigms to predict nonseparable Hamiltonian systems that are
ubiquitous in fluid dynamics and quantum mechanics were rarely explored. The
main computational challenge lies in the effective embedding of symplectic
priors to describe the inherently coupled evolution of position and momentum,
which typically exhibits intricate dynamics. To solve the problem, we propose a
novel neural network architecture, Nonseparable Symplectic Neural Networks
(NSSNNs), to uncover and embed the symplectic structure of a nonseparable
Hamiltonian system from limited observation data. The enabling mechanics of our
approach is an augmented symplectic time integrator to decouple the position
and momentum energy terms and facilitate their evolution. We demonstrated the
efficacy and versatility of our method by predicting a wide range of
Hamiltonian systems, both separable and nonseparable, including chaotic
vortical flows. We showed the unique computational merits of our approach to
yield long-term, accurate, and robust predictions for large-scale Hamiltonian
systems by rigorously enforcing symplectomorphism.
