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A structure-preserving kernel method for learning Hamiltonian systems.

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Title: A structure-preserving kernel method for learning Hamiltonian systems.
Authors: Hu, Jianyu1 (AUTHOR), Ortega, Juan-Pablo1 (AUTHOR), Yin, Daiying1 (AUTHOR)
Source: Mathematics of Computation. Jul2026, Vol. 95 Issue 360, p1719-1774. 56p.
Subjects: Hamiltonian systems, Reproducing kernel (Mathematics), Hamilton's principle function, Error analysis in mathematics, Numerical analysis, Loss functions (Statistics), Ridge regression (Statistics)
Abstract: A structure-preserving kernel ridge regression method is presented that allows the recovery of 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 together with the convergence rate is illustrated with various numerical experiments. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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Header DbId: egs
DbLabel: Engineering Source
An: 192903048
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
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  Data: A structure-preserving kernel method for learning Hamiltonian systems.
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Computation%22">Mathematics of Computation</searchLink>. Jul2026, Vol. 95 Issue 360, p1719-1774. 56p.
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  Data: <searchLink fieldCode="DE" term="%22Hamiltonian+systems%22">Hamiltonian systems</searchLink><br /><searchLink fieldCode="DE" term="%22Reproducing+kernel+%28Mathematics%29%22">Reproducing kernel (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Hamilton's+principle+function%22">Hamilton's principle function</searchLink><br /><searchLink fieldCode="DE" term="%22Error+analysis+in+mathematics%22">Error analysis in mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Loss+functions+%28Statistics%29%22">Loss functions (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Ridge+regression+%28Statistics%29%22">Ridge regression (Statistics)</searchLink>
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  Label: Abstract
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  Data: A structure-preserving kernel ridge regression method is presented that allows the recovery of 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 together with the convergence rate is illustrated with various numerical experiments. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1090/mcom/4106
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 56
        StartPage: 1719
    Subjects:
      – SubjectFull: Hamiltonian systems
        Type: general
      – SubjectFull: Reproducing kernel (Mathematics)
        Type: general
      – SubjectFull: Hamilton's principle function
        Type: general
      – SubjectFull: Error analysis in mathematics
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Loss functions (Statistics)
        Type: general
      – SubjectFull: Ridge regression (Statistics)
        Type: general
    Titles:
      – TitleFull: A structure-preserving kernel method for learning Hamiltonian systems.
        Type: main
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          Name:
            NameFull: Hu, Jianyu
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            NameFull: Ortega, Juan-Pablo
      – PersonEntity:
          Name:
            NameFull: Yin, Daiying
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      – BibEntity:
          Dates:
            – D: 01
              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
          Identifiers:
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              Value: 00255718
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              Value: 95
            – Type: issue
              Value: 360
          Titles:
            – TitleFull: Mathematics of Computation
              Type: main
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