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

Symplectic Gaussian Process Regression of Hamiltonian Flow Maps