Symplecticity-Preserving Prediction of Hamiltonian Dynamics by Generalized Kernel Interpolation

TL;DR

This paper introduces a kernel-based surrogate method for integrating Hamiltonian dynamics that preserves symplectic structure exactly, enabling accurate long-term predictions for complex physical systems.

Contribution

It proposes a novel symplectic kernel predictor that learns a scalar potential and uses a gradient Hermite--Birkhoff interpolation in RKHS, ensuring structure-preserving integration.

Findings

Achieves nearly algebraic greedy convergence.

Reduces long-time trajectory errors by 2-3 orders of magnitude.

Effective for high-dimensional PDEs with structure-preserving model reduction.

Abstract

In this work, a kernel-based surrogate for integrating Hamiltonian dynamics that is symplectic by construction and tailored to large prediction horizons is proposed. The method learns a scalar potential whose gradient enters a symplectic-Euler update, yielding a discrete flow map that exactly preserves the canonical symplectic structure. Training is formulated as a gradient Hermite--Birkhoff interpolation problem in a reproducing kernel Hilbert space, providing a systematic framework for existence, uniqueness, and error control. Algorithmically, the symplectic kernel predictor is combined with structure-preserving model order reduction, enabling efficient treatment of high-dimensional discretized PDEs. Numerical tests for a pendulum, a nonlinear spring--mass chain, and a semi-discrete wave equation show nearly algebraic greedy convergence and long-time trajectory errors reduce by two to three orders of magnitude compared to an implicit midpoint baseline at the same macro time step.