Symplectic Gaussian Process Regression of Hamiltonian Flow Maps

TL;DR

This paper introduces a symplectic Gaussian process regression method for modeling Hamiltonian flow maps, enabling accurate long-term simulations and Hamiltonian learning from data, with improved performance over existing methods.

Contribution

The paper develops a novel symplectic Gaussian process regression framework that enforces symplectic structure, providing accurate implicit and explicit methods for Hamiltonian systems from scattered data.

Findings

Methods accurately model Hamiltonian flow maps.

Symplectic Gaussian process regression outperforms existing approaches.

Able to learn Hamiltonian functions from observed data.

Abstract

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