{"type":"journal_article","status":"public","project":[{"name":"PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing","_id":"52"}],"_id":"53805","user_id":"85279","department":[{"_id":"636"}],"article_type":"original","file_date_updated":"2025-05-02T13:20:31Z","publication_status":"epub_ahead","has_accepted_license":"1","related_material":{"link":[{"relation":"software","description":"GitHub","url":"https://github.com/Christian-Offen/Lagrangian_GP"}]},"citation":{"ama":"Offen C. Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification. <i>Mathematics of Computation</i>. Published online 2025. doi:<a href=\"https://doi.org/10.1090/mcom/4120\">10.1090/mcom/4120</a>","chicago":"Offen, Christian. “Machine Learning of Continuous and Discrete Variational ODEs with Convergence Guarantee and Uncertainty Quantification.” <i>Mathematics of Computation</i>, 2025. <a href=\"https://doi.org/10.1090/mcom/4120\">https://doi.org/10.1090/mcom/4120</a>.","ieee":"C. Offen, “Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification,” <i>Mathematics of Computation</i>, 2025, doi: <a href=\"https://doi.org/10.1090/mcom/4120\">10.1090/mcom/4120</a>.","apa":"Offen, C. (2025). Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification. <i>Mathematics of Computation</i>. <a href=\"https://doi.org/10.1090/mcom/4120\">https://doi.org/10.1090/mcom/4120</a>","short":"C. Offen, Mathematics of Computation (2025).","bibtex":"@article{Offen_2025, title={Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification}, DOI={<a href=\"https://doi.org/10.1090/mcom/4120\">10.1090/mcom/4120</a>}, journal={Mathematics of Computation}, publisher={American Mathematical Society}, author={Offen, Christian}, year={2025} }","mla":"Offen, Christian. “Machine Learning of Continuous and Discrete Variational ODEs with Convergence Guarantee and Uncertainty Quantification.” <i>Mathematics of Computation</i>, American Mathematical Society, 2025, doi:<a href=\"https://doi.org/10.1090/mcom/4120\">10.1090/mcom/4120</a>."},"date_updated":"2025-06-29T13:03:55Z","oa":"1","author":[{"orcid":"0000-0002-5940-8057","last_name":"Offen","id":"85279","full_name":"Offen, Christian","first_name":"Christian"}],"doi":"10.1090/mcom/4120","publication":"Mathematics of Computation","abstract":[{"text":"The article introduces a method to learn dynamical systems that are governed by Euler–Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.","lang":"eng"}],"file":[{"content_type":"application/pdf","relation":"main_file","creator":"coffen","date_created":"2025-05-02T13:20:31Z","date_updated":"2025-05-02T13:20:31Z","file_id":"59759","file_name":"L_Collocation_ODE_mcom-l-template.pdf","access_level":"open_access","title":"Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification ","file_size":1819189,"description":"The article introduces a method to learn dynamical systems that\nare governed by Euler–Lagrange equations from data. The method is based on\nGaussian process regression and identifies continuous or discrete Lagrangians\nand is, therefore, structure preserving by design. A rigorous proof of con-\nvergence as the distance between observation data points converges to zero\nand lower bounds for convergence rates are provided. Next to convergence\nguarantees, the method allows for quantification of model uncertainty, which\ncan provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian,\nincluding of Hamiltonian functions (energy) and symplectic structures, which\nis of interest in the context of system identification. The article overcomes\nmajor practical and theoretical difficulties related to the ill-posedness of the\nidentification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex\nminimisation problems in reproducing kernel Hilbert spaces."}],"external_id":{"arxiv":["arXiv:2404.19626"]},"ddc":["510"],"language":[{"iso":"eng"}],"quality_controlled":"1","year":"2025","publisher":"American Mathematical Society","date_created":"2024-04-30T16:04:40Z","title":"Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification"}