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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 together with the convergence rate is illustrated with various numerical experiments.<\/p>","DOI":"10.1090\/mcom\/4106","type":"journal-article","created":{"date-parts":[[2025,6,16]],"date-time":"2025-06-16T14:24:07Z","timestamp":1750083847000},"page":"1719-1774","source":"Crossref","is-referenced-by-count":6,"title":["A structure-preserving kernel method for learning Hamiltonian systems"],"prefix":"10.1090","volume":"95","author":[{"given":"Jianyu","family":"Hu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Juan-Pablo","family":"Ortega","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daiying","family":"Yin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,6,16]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[Abra~08] R. 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