  We consider the learning and prediction of nonlinear time series generated by
a latent symplectic map. A special case is (not necessarily separable)
Hamiltonian systems, whose solution flows give such symplectic maps. For this
special case, both generic approaches based on learning the vector field of the
latent ODE and specialized approaches based on learning the Hamiltonian that
generates the vector field exist. Our method, however, is different as it does
not rely on the vector field nor assume its existence; instead, it directly
learns the symplectic evolution map in discrete time. Moreover, we do so by
representing the symplectic map via a generating function, which we approximate
by a neural network (hence the name GFNN). This way, our approximation of the
evolution map is always \emph{exactly} symplectic. This additional geometric
structure allows the local prediction error at each step to accumulate in a
controlled fashion, and we will prove, under reasonable assumptions, that the
global prediction error grows at most \emph{linearly} with long prediction
time, which significantly improves an otherwise exponential growth. In
addition, as a map-based and thus purely data-driven method, GFNN avoids two
additional sources of inaccuracies common in vector-field based approaches,
namely the error in approximating the vector field by finite difference of the
data, and the error in numerical integration of the vector field for making
predictions. Numerical experiments further demonstrate our claims.
