Data-efficient Kernel Methods for Learning Hamiltonian Systems

TL;DR

This paper introduces kernel-based methods for data-efficient learning and forecasting of Hamiltonian systems, ensuring accurate predictions and conservation properties, with theoretical error guarantees and broader applicability.

Contribution

It proposes novel one-step and two-step kernel methods for Hamiltonian system identification, outperforming baselines especially with limited data, and offers a general framework for arbitrary dynamical systems.

Findings

Accurate, data-efficient predictions on benchmark systems

Outperforms two-step kernel baselines in scarce-data regimes

Provides theoretical error estimates for learned models

Abstract

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from data. We present two approaches: a two-step method that reconstructs trajectories before learning the Hamiltonian, and a one-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms two-step kernel-based baselines, particularly in scarce-data regimes, while preserving the conservation properties of Hamiltonian dynamics. Moreover, our methodology provides theoretical a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.