{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T01:13:41Z","timestamp":1784855621805,"version":"3.55.0"},"reference-count":58,"publisher":"American Mathematical Society (AMS)","issue":"361","license":[{"start":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T00:00:00Z","timestamp":1782432000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100014690","name":"Ministerium f\u00fcr Kultur und Wissenschaft des Landes Nordrhein-Westfalen","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100014690","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The article introduces a method to learn dynamical systems that are governed by Euler\u2013Lagrange 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.<\/p>","DOI":"10.1090\/mcom\/4120","type":"journal-article","created":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T10:28:47Z","timestamp":1750933727000},"page":"2339-2386","source":"Crossref","is-referenced-by-count":4,"title":["Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification"],"prefix":"10.1090","volume":"95","author":[{"given":"Christian","family":"Offen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,6,26]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (Amsterdam)","isbn-type":"print","volume-title":"Sobolev spaces","volume":"140","author":"Adams, Robert A.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0120441438","edition":"2"},{"key":"2","unstructured":"T. Aoshima, T. Matsubara, and T. Yaguchi, Deep discrete-time Lagrangian mechanics, ICLR SimDL, vol. 49, 2021, pp. 1\u20138."},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s00211-007-0092-z","article-title":"An extension of a bound for functions in Sobolev spaces, with applications to (\ud835\udc5a,\ud835\udc60)-spline interpolation and smoothing","volume":"107","author":"Arcang\u00e9li, R\u00e9mi","year":"2007","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Springer New York, New York, NY, 1989, DOI 10.1007\/978-1-4757-2063-1.","DOI":"10.1007\/978-1-4757-2063-1"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1145\/355694.355697","article-title":"A comparison of algorithms for solving symmetric indefinite systems of linear equations","volume":"2","author":"Barwell, Victor","year":"1976","journal-title":"ACM Trans. Math. Software","ISSN":"https:\/\/id.crossref.org\/issn\/0098-3500","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"Paper No. 113488, 23","DOI":"10.1016\/j.jcp.2024.113488","article-title":"Error analysis of kernel\/GP methods for nonlinear and parametric PDEs","volume":"520","author":"Batlle, Pau","year":"2025","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"12","key":"7","doi-asserted-by":"publisher","first-page":"121107","DOI":"10.1063\/1.5128231","article-title":"On learning Hamiltonian systems from data","volume":"29","author":"Bertalan, Tom","year":"2019","journal-title":"Chaos","ISSN":"https:\/\/id.crossref.org\/issn\/1054-1500","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","article-title":"Julia: a fresh approach to numerical computing","volume":"59","author":"Bezanson, Jeff","year":"2017","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"137","key":"9","doi-asserted-by":"publisher","first-page":"163","DOI":"10.2307\/2005787","article-title":"Some stable methods for calculating inertia and solving symmetric linear systems","volume":"31","author":"Bunch, James R.","year":"1977","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/0305-4470\/16\/1\/010","article-title":"Non-Noether constants of motion","volume":"16","author":"Cari\u00f1ena, Jos\u00e9 F.","year":"1983","journal-title":"J. Phys. A","ISSN":"https:\/\/id.crossref.org\/issn\/0305-4470","issn-type":"print"},{"key":"11","unstructured":"R. Chen and M. Tao, Data-driven prediction of general Hamiltonian dynamics via learning exactly-symplectic maps, Proceedings of the 38th International Conference on Machine Learning, Proceedings of Machine Learning Research, vol. 139, PMLR, 18\u201324 July 2021, pp. 1717\u20131727."},{"key":"12","doi-asserted-by":"publisher","first-page":"Paper No. 110668, 29","DOI":"10.1016\/j.jcp.2021.110668","article-title":"Solving and learning nonlinear PDEs with Gaussian processes","volume":"447","author":"Chen, Yifan","year":"2021","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"13","unstructured":"Y. Chen, B. Xu, T. Matsubara, and T. Yaguchi, Variational principle and variational integrators for neural symplectic forms, ICML Workshop on New Frontiers in Learning, Control, and Dynamical Systems, 2023."},{"key":"14","doi-asserted-by":"crossref","unstructured":"A. Christmann and I. Steinwart, Kernels and reproducing kernel Hilbert spaces, Springer New York, New York, NY, 2008, pp. 110\u2013163.","DOI":"10.1007\/978-0-387-77242-4_4"},{"key":"15","unstructured":"N. Da Costa, M. Pf\u00f6rtner, L. Da Costa, and P. Hennig, Sample path regularity of Gaussian processes from the covariance kernel, 2024,  arXiv:2312.14886."},{"key":"16","unstructured":"M. Cranmer, S. Greydanus, S. Hoyer, P. Battaglia, D. Spergel, and S. Ho, Lagrangian neural networks, 2020,  arXiv:2003.04630."},{"key":"17","doi-asserted-by":"publisher","first-page":"Paper No. 112495, 13","DOI":"10.1016\/j.jcp.2023.112495","article-title":"Symplectic learning for Hamiltonian neural networks","volume":"494","author":"David, Marco","year":"2023","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"18","doi-asserted-by":"publisher","first-page":"Paper No. 063115, 12","DOI":"10.1063\/5.0142969","article-title":"Hamiltonian neural networks with automatic symmetry detection","volume":"33","author":"Dierkes, Eva","year":"2023","journal-title":"Chaos","ISSN":"https:\/\/id.crossref.org\/issn\/1054-1500","issn-type":"print"},{"key":"19","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5200-9","volume-title":"Sequences and series in Banach spaces","volume":"92","author":"Diestel, Joseph","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0387908595"},{"key":"20","series-title":"Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics","isbn-type":"print","volume-title":"Multivariate statistics","author":"Eaton, Morris L.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0471027766"},{"key":"21","doi-asserted-by":"publisher","first-page":"Paper No. 112738, 25","DOI":"10.1016\/j.jcp.2023.112738","article-title":"Pseudo-Hamiltonian neural networks for learning partial differential equations","volume":"500","author":"Eidnes, S\u00f8lve","year":"2024","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"22","doi-asserted-by":"crossref","unstructured":"G. Evangelisti and S. Hirche, Physically consistent learning of conservative Lagrangian systems with Gaussian processes, 2022 IEEE 61st Conference on Decision and Control (CDC), IEEE, 2022.","DOI":"10.1109\/CDC51059.2022.9993123"},{"issue":"349","key":"23","doi-asserted-by":"publisher","first-page":"2391","DOI":"10.1090\/mcom\/3915","article-title":"Learning particle swarming models from data with Gaussian processes","volume":"93","author":"Feng, Jinchao","year":"2024","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"24","unstructured":"I. M. Gelfand, S. V. Fomin, and R. A. Silverman, Calculus of variations, Dover Books on Mathematics, Dover Publications, 2000."},{"key":"25","unstructured":"S. Greydanus, M. Dzamba, and J. Yosinski, Hamiltonian neural networks, Advances in Neural Information Processing Systems, vol. 32, Curran Associates, Inc., 2019."},{"issue":"1","key":"26","doi-asserted-by":"publisher","first-page":"Research Paper 1, 13","DOI":"10.37236\/1027","article-title":"Combinatorics of partial derivatives","volume":"13","author":"Hardy, Michael","year":"2006","journal-title":"Electron. J. Combin."},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/0003-4916(82)90334-7","article-title":"Equations of motion, commutation relations and ambiguities in the Lagrangian formalism","volume":"140","author":"Henneaux, Marc","year":"1982","journal-title":"Ann. Physics","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4916","issn-type":"print"},{"key":"28","doi-asserted-by":"crossref","unstructured":"J. Hu, J.-P. Ortega, and D. Yin, A structure-preserving kernel method for learning Hamiltonian systems, Mathematics of Computation, 2025, DOI 10.1090\/mcom\/4106.","DOI":"10.1090\/mcom\/4106"},{"key":"29","doi-asserted-by":"crossref","unstructured":"P. Jin, Z. Zhang, A. Zhu, Y. Tang, and G. E. Karniadakis, SympNets: intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems, Neural Net. 132 (2020), 166\u2013179.","DOI":"10.1016\/j.neunet.2020.08.017"},{"key":"30","doi-asserted-by":"crossref","unstructured":"Y. Lishkova, P. Scherer, S. Ridderbusch, M. Jamnik, P. Li\u00f2, S. Ober-Bl\u00f6baum, and C. Offen, Discrete Lagrangian neural networks with automatic symmetry discovery, IFAC-PapersOnLine 56 (2023), no. 2, 3203\u20133210, 22nd IFAC World Congress.","DOI":"10.1016\/j.ifacol.2023.10.1457"},{"issue":"29","key":"31","doi-asserted-by":"publisher","first-page":"14424","DOI":"10.1073\/pnas.1822012116","article-title":"Nonparametric inference of interaction laws in systems of agents from trajectory data","volume":"116","author":"Lu, Fei","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA","ISSN":"https:\/\/id.crossref.org\/issn\/0027-8424","issn-type":"print"},{"issue":"3","key":"32","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/0034-4877(89)90071-2","article-title":"On the inverse problem with symmetries, and the appearance of cohomologies in classical Lagrangian dynamics","volume":"28","author":"Marmo, G.","year":"1989","journal-title":"Rep. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0034-4877","issn-type":"print"},{"key":"33","doi-asserted-by":"crossref","unstructured":"G. Marmo and C. Rubano, On the uniqueness of the Lagrangian description for charged particles in external magnetic field, Il Nuovo Cimento A 98 (1987), no. 4, 387\u2013399.","DOI":"10.1007\/BF02902083"},{"key":"34","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21792-5","volume-title":"Introduction to mechanics and symmetry","volume":"17","author":"Marsden, Jerrold E.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/038798643X","edition":"2"},{"key":"35","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1017\/S096249290100006X","article-title":"Discrete mechanics and variational integrators","volume":"10","author":"Marsden, J. E.","year":"2001","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"250","key":"36","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1090\/S0025-5718-04-01708-9","article-title":"Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting","volume":"74","author":"Narcowich, Francis J.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"37","doi-asserted-by":"publisher","first-page":"Paper No. 114780, 18","DOI":"10.1016\/j.cam.2022.114780","article-title":"Variational learning of Euler-Lagrange dynamics from data","volume":"421","author":"Ober-Bl\u00f6baum, Sina","year":"2023","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"38","doi-asserted-by":"crossref","unstructured":"C. Offen, Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification, 2024,  arXiv:2407.07642.","DOI":"10.1090\/mcom\/4120"},{"key":"39","unstructured":"C. Offen, Software: Christian-Offen\/Lagrangian_GP: initial release of GitHub repository, 2024, DOI 10.5281\/zenodo.11093645."},{"issue":"1","key":"40","doi-asserted-by":"publisher","first-page":"Paper No. 013122, 13","DOI":"10.1063\/5.0065913","article-title":"Symplectic integration of learned Hamiltonian systems","volume":"32","author":"Offen, C.","year":"2022","journal-title":"Chaos","ISSN":"https:\/\/id.crossref.org\/issn\/1054-1500","issn-type":"print"},{"key":"41","isbn-type":"print","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1007\/978-3-031-38271-0_57","article-title":"Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves","author":"Offen, Christian","year":"[2023] \\copyright2023","ISBN":"https:\/\/id.crossref.org\/isbn\/9783031382703"},{"key":"42","doi-asserted-by":"crossref","unstructured":"C. Offen and S. Ober-Bl\u00f6baum, Learning of discrete models of variational PDEs from data, Chaos 34 (2024), 013104.","DOI":"10.1063\/5.0172287"},{"key":"43","first-page":"Paper No. [68], 56","article-title":"Learnability of linear port-Hamiltonian systems","volume":"25","author":"Ortega, Juan-Pablo","year":"2024","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"44","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1017\/9781108594967","volume-title":"Operator-adapted wavelets, fast solvers, and numerical homogenization","volume":"35","author":"Owhadi, Houman","year":"2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9781108484367"},{"key":"45","unstructured":"M. Pf\u00f6rtner, I. Steinwart, P. Hennig, and J. Wenger, Physics-informed Gaussian process regression generalizes linear PDE solvers, 2024,  arXiv:2212.12474."},{"key":"46","doi-asserted-by":"crossref","unstructured":"H. Qin, Machine learning and serving of discrete field theories, Sci. Rep. 10 (2020), no. 1, 19329.","DOI":"10.1038\/s41598-020-76301-0"},{"key":"47","first-page":"1939","article-title":"A unifying view of sparse approximate Gaussian process regression","volume":"6","author":"Qui\u00f1onero-Candela, Joaquin","year":"2005","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"48","doi-asserted-by":"crossref","unstructured":"C. Rackauckas and Q. Nie, Differentialequations.jl\u2013a performant and feature-rich ecosystem for solving differential equations in Julia, J. Open Res. Softw. 5 (2017), no. 1, 15.","DOI":"10.5334\/jors.151"},{"key":"49","series-title":"Adaptive Computation and Machine Learning","isbn-type":"print","volume-title":"Gaussian processes for machine learning","author":"Rasmussen, Carl Edward","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780262182539"},{"issue":"5","key":"50","doi-asserted-by":"publisher","first-page":"Paper No. 053121, 24","DOI":"10.1063\/5.0048129","article-title":"Symplectic Gaussian process regression of maps in Hamiltonian systems","volume":"31","author":"Rath, Katharina","year":"2021","journal-title":"Chaos","ISSN":"https:\/\/id.crossref.org\/issn\/1054-1500","issn-type":"print"},{"key":"51","doi-asserted-by":"crossref","unstructured":"T. Roub\u00ed\u010dek, Calculus of Variations, John Wiley & Sons Ltd., 2015, pp. 1\u201338.","DOI":"10.1002\/3527600434.eap735"},{"key":"52","series-title":"International Series in Pure and Applied Mathematics","isbn-type":"print","volume-title":"Functional analysis","author":"Rudin, Walter","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0070542368","edition":"2"},{"key":"53","isbn-type":"print","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1017\/S0962492906270016","article-title":"Kernel techniques: from machine learning to meshless methods","volume":"15","author":"Schaback, Robert","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0521868157","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"3","key":"54","doi-asserted-by":"publisher","first-page":"A2019--A2046","DOI":"10.1137\/20M1336254","article-title":"Sparse Cholesky factorization by Kullback-Leibler minimization","volume":"43","author":"Sch\u00e4fer, Florian","year":"2021","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"55","doi-asserted-by":"publisher","first-page":"688","DOI":"10.1137\/19M129526X","article-title":"Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity","volume":"19","author":"Sch\u00e4fer, Florian","year":"2021","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"key":"56","unstructured":"M. Titsias, Variational learning of inducing variables in sparse Gaussian processes, Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics (Hilton Clearwater Beach Resort, Clearwater Beach, Florida), Proceedings of Machine Learning Research, vol. 5, PMLR, 16\u201318 April 2009, pp. 567\u2013574."},{"issue":"4","key":"57","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1007\/s00211-017-0896-4","article-title":"Modified equations for variational integrators","volume":"137","author":"Vermeeren, Mats","year":"2017","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"58","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1007\/s00211-005-0637-y","article-title":"Approximate interpolation with applications to selecting smoothing parameters","volume":"101","author":"Wendland, Holger","year":"2005","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04120-6\/mcom4120_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04120-6\/S0025-5718-2025-04120-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:53:43Z","timestamp":1781286823000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04120-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,26]]},"references-count":58,"journal-issue":{"issue":"361","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["S0025-5718-2025-04120-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4120","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2025,6,26]]}}}