  Hamiltonian systems are differential equations which describe systems in
classical mechanics, plasma physics, and sampling problems. They exhibit many
structural properties, such as a lack of attractors and the presence of
conservation laws. To predict Hamiltonian dynamics based on discrete trajectory
observations, incorporation of prior knowledge about Hamiltonian structure
greatly improves predictions. This is typically done by learning the system's
Hamiltonian and then integrating the Hamiltonian vector field with a symplectic
integrator. For this, however, Hamiltonian data needs to be approximated based
on the trajectory observations. Moreover, the numerical integrator introduces
an additional discretisation error. In this paper, we show that an inverse
modified Hamiltonian structure adapted to the geometric integrator can be
learned directly from observations. A separate approximation step for the
Hamiltonian data avoided. The inverse modified data compensates for the
discretisation error such that the discretisation error is eliminated. The
technique is developed for Gaussian Processes.

Numerical Analysis;Numerical Analysis