A structure-preserving kernel ridge regression method is presented that allows the recovery of potentially high-dimensional and nonlinear Hamiltonian functions out of datasets made of noisy observations of Hamiltonian vector fields. The method proposes a closed-form solution that yields excellent numerical performances that surpass other techniques proposed in the literature in this setup. From the methodological point of view, the paper extends kernel regression methods to problems in which loss functions involving linear functions of gradients are required and, in particular, a differential reproducing property and a Representer Theorem are proved in this context. The relation between the structure-preserving kernel estimator and the Gaussian posterior mean estimator is analyzed. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator is illustrated with various numerical experiments.

给出一种结构保持的核岭回归方法，可以从由哈密顿矢量场的噪声观测组成的数据集中恢复可能是高维和非线性的哈密顿函数。该方法提供了一个闭式解，其数值性能优于文献中提出的其他技术。从方法论的角度，本文将核回归方法扩展到需要涉及梯度线性函数的损失函数的问题，特别是在这种背景下证明了微分再生性质和Representer定理。分析了结构保持的核估计量与高斯后验均值估计量之间的关系。进行了完整的误差分析，使用固定和自适应正则化参数提供了收敛速度。通过各种数值实验展示了所提出估计器的良好性能。

一种保结构核方法用于学习哈密顿系统