Data-efficient Kernel Methods for Learning Hamiltonian Systems
This work addresses the challenge of simulating physical systems for scientific and engineering applications, offering an incremental improvement with a more efficient kernel-based approach.
The authors tackled the problem of data-driven simulation of Hamiltonian systems by proposing kernel-based methods for identification and forecasting, achieving accurate and data-efficient predictions that outperform baselines in scarce-data regimes while preserving conservation properties.
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.