Symplectic integration of learned Hamiltonian systems

TL;DR

This paper introduces a method to directly learn an inverse modified Hamiltonian structure from trajectory observations, effectively compensating for discretisation errors in symplectic integration of Hamiltonian systems, using Gaussian Processes.

Contribution

It proposes a novel approach to learn inverse modified Hamiltonians directly from data, bypassing Hamiltonian approximation and reducing discretisation errors in symplectic integrators.

Findings

The method accurately predicts Hamiltonian dynamics from discrete observations.

It effectively compensates for numerical discretisation errors.

The approach is demonstrated using Gaussian Processes.

Abstract

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.