Data-efficient Kernel Methods for Learning Hamiltonian Systems

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 trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-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 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove 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.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Dynamical Systems (math.DS); Machine Learning (stat.ML)

arXiv:2509.17154 [math.NA]

(or arXiv:2509.17154v2 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2509.17154 Focus to learn more arXiv-issued DOI via DataCite

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