{"ID":2868196,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.17154","arxiv_id":"2509.17154","title":"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 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.","short_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 approac...","url_abs":"https://arxiv.org/abs/2509.17154","url_pdf":"https://arxiv.org/pdf/2509.17154v1","authors":"[\"Yasamin Jalalian\",\"Mostafa Samir\",\"Boumediene Hamzi\",\"Peyman Tavallali\",\"Houman Owhadi\"]","published":"2025-09-21T16:50:17Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.LG\",\"math.DS\",\"stat.ML\"]","methods":"[]","has_code":false}