Learning Hamiltonians from Trajectory Data with Kernel Methods
Abstract
We study the recovery of a Hamiltonian from discrete trajectory observations using structure-preserving kernel methods. When exact-flow data are fitted through a symplectic one-step scheme, the fixed-step learning target is generally not the physical Hamiltonian itself, but a scheme-dependent inverse modified Hamiltonian. For the implicit midpoint and p-implicit symplectic Euler methods, we establish the existence of analytic inverse targets and show that their discrepancies from the physical Hamiltonian are second and first order in the observation step, respectively. Within an RKHS framework, we derive finite-sample reconstruction bounds for random trajectories, allowing dependence among observations from the same trajectory, and separate the error into estimation, approximation, and integrator-induced components. We further establish stability under noisy trajectory observations and develop a multistride Richardson correction that systematically raises the order of the integrator-induced error without requiring additional trajectories. Numerical experiments confirm the predicted scheme-dependent orders, the effect of Richardson correction, and the applicability of the method across several Hamiltonian systems.
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BibTeX
@misc{hu2026learning,
title = {{Learning Hamiltonians from Trajectory Data with Kernel Methods}},
author = {Jianyu Hu and David Martin de Diego and Juan-Pablo Ortega and Miguel Vaquero},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17835},
eprint = {2610.17835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17835},
}
RIS
TY - UNPB
TI - Learning Hamiltonians from Trajectory Data with Kernel Methods
AU - Jianyu Hu
AU - David Martin de Diego
AU - Juan-Pablo Ortega
AU - Miguel Vaquero
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17835
UR - https://arxiv.org/abs/2610.17835
AB - We study the recovery of a Hamiltonian from discrete trajectory observations using structure-preserving kernel methods. When exact-flow data are fitted through a symplectic one-step scheme, the fixed-step learning target is generally not the physical Hamiltonian itself, but a scheme-dependent inverse modified Hamiltonian. For the implicit midpoint and p-implicit symplectic Euler methods, we establish the existence of analytic inverse targets and show that their discrepancies from the physical Hamiltonian are second and first order in the observation step, respectively. Within an RKHS framework, we derive finite-sample reconstruction bounds for random trajectories, allowing dependence among observations from the same trajectory, and separate the error into estimation, approximation, and integrator-induced components. We further establish stability under noisy trajectory observations and develop a multistride Richardson correction that systematically raises the order of the integrator-induced error without requiring additional trajectories. Numerical experiments confirm the predicted scheme-dependent orders, the effect of Richardson correction, and the applicability of the method across several Hamiltonian systems.
ER -