Symplectic integration of learned Hamiltonian systems

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.

References in corpus (5)

• Symplectic Learning for Hamiltonian Neural Networks
• Symplectic Gaussian Process Regression of Hamiltonian Flow Maps
• Inverse modified differential equations for discovery of dynamics
• Learning ODE Models with Qualitative Structure Using Gaussian Processes
• Bifurcation preserving discretisations of optimal control problems

Cited by in corpus (5)

• Variational Learning of Euler-Lagrange Dynamics from Data
• Hamiltonian Neural Networks with Automatic Symmetry Detection
• Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery
• Backward error analysis for variational discretisations of partial differential equations
• Backward error analysis for conjugate symplectic methods