用核方法高效学习哈密顿系统，数据少也能准预测。
Data-efficient Kernel Methods for Learning Hamiltonian Systems
- 分两步或一步直接从数据中学习哈密顿量
- 在少量数据下仍保持高精度，优于现有方法
- 适合物理建模与动力系统研究者使用
哈密顿动力学描述了广泛的物理系统，因此基于数据的哈密顿系统模拟对众多科学和工程问题至关重要。本文提出基于核的方法，直接从数据中识别并预测哈密顿系统。我们设计两种策略：先重构轨迹再学习哈密顿量的两步法，以及联合推断轨迹与哈密顿量的一步法。在质量-弹簧系统、非线性摆及Henon-Heiles系统等多个基准测试中，本框架在数据稀缺情况下仍实现高精度预测，显著优于两步核基基线，同时保持哈密顿系统的守恒性质。此外，方法提供理论上的先验误差估计，确保模型可靠性。我们还构建了一个更通用的问题无关数值框架，可拓展至任意动力系统的数据驱动学习。
原文摘要 · Abstract (English)
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.
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