  We present an approach to construct appropriate and efficient emulators for
Hamiltonian flow maps. Intended future applications are long-term tracing of
fast charged particles in accelerators and magnetic plasma confinement
configurations. The method is based on multi-output Gaussian process regression
on scattered training data. To obtain long-term stability the symplectic
property is enforced via the choice of the matrix-valued covariance function.
Based on earlier work on spline interpolation we observe derivatives of the
generating function of a canonical transformation. A product kernel produces an
accurate implicit method, whereas a sum kernel results in a fast explicit
method from this approach. Both correspond to a symplectic Euler method in
terms of numerical integration. These methods are applied to the pendulum and
the H\'enon-Heiles system and results compared to an symmetric regression with
orthogonal polynomials. In the limit of small mapping times, the Hamiltonian
function can be identified with a part of the generating function and thereby
learned from observed time-series data of the system's evolution. Besides
comparable performance of implicit kernel and spectral regression for
symplectic maps, we demonstrate a substantial increase in performance for
learning the Hamiltonian function compared to existing approaches.
