Symplectic integration of learned Hamiltonian systems

Hamiltonian systems are differential equations that 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, the 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 need to be approximated based on trajectory observations. Moreover, the numerical integrator introduces an additional discretization error. In this article, 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 is avoided. The inverse modified data compensate for the discretization error such that the discretization error is eliminated. The technique is developed for Gaussian processes.