Data-efficient Kernel Methods for Learning Hamiltonian Systems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916959827787776 |
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| author | Jalalian, Yasamin Samir, Mostafa Hamzi, Boumediene Tavallali, Peyman Owhadi, Houman |
| author_facet | Jalalian, Yasamin Samir, Mostafa Hamzi, Boumediene Tavallali, Peyman Owhadi, Houman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Data-efficient Kernel Methods for Learning Hamiltonian Systems Jalalian, Yasamin Samir, Mostafa Hamzi, Boumediene Tavallali, Peyman Owhadi, Houman Numerical Analysis Machine Learning Dynamical Systems 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. |
| title | Data-efficient Kernel Methods for Learning Hamiltonian Systems |
| topic | Numerical Analysis Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2509.17154 |