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.

翻译：本文提出了一种保持结构的核岭回归方法，该方法能够从哈密顿向量场的含噪观测数据中恢复潜在高维非线性哈密顿函数。该方法采用闭式解，其数值表现优异，超越了文献中其他现有技术。从方法论角度而言，本文将核回归方法扩展至需要包含梯度线性函数的损失函数的问题，并在此背景下证明了微分再生性质与表示定理。同时分析了保持结构核估计量与高斯后验均值估计量之间的关联。通过固定与自适应正则化参数，开展了完整的误差分析并给出了收敛速率。多种数值实验验证了所提估计量的良好性能。