[{"year":"2023","quality_controlled":"1","title":"Variational Learning of Euler–Lagrange Dynamics from Data","date_created":"2022-01-11T13:24:00Z","publisher":"Elsevier","file":[{"file_id":"32274","access_level":"open_access","description":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions\nconnecting two points in a phase space, the system will exhibit those motions which extremise an action functional.\nMany qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equa-\ntions, are related to the existence of an action functional. Incorporating variational structure into learning algorithms\nfor dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features\nwith the exact physical system. In this paper we show how to incorporate variational principles into trajectory predic-\ntions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position\ndata of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no\nprior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward\nerror analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the\nlearned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of\nvariational backward error analysis. (3) Finally, we introduce a method to perform system identification from position\nobservations only, based on variational backward error analysis.","title":"Variational Learning of Euler–Lagrange Dynamics from Data","date_created":"2022-06-28T15:25:50Z","date_updated":"2022-06-28T15:25:50Z","relation":"main_file","file_name":"ShadowLagrangian_revision1_journal_style_arxiv.pdf","file_size":3640770,"creator":"coffen","content_type":"application/pdf"}],"abstract":[{"text":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\r\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.","lang":"eng"}],"publication":"Journal of Computational and Applied Mathematics","language":[{"iso":"eng"}],"keyword":["Lagrangian learning","variational backward error analysis","modified Lagrangian","variational integrators","physics informed learning"],"ddc":["510"],"external_id":{"arxiv":["2112.12619"]},"intvolume":"       421","page":"114780","citation":{"apa":"Ober-Blöbaum, S., &#38; Offen, C. (2023). Variational Learning of Euler–Lagrange Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>, <i>421</i>, 114780. <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">https://doi.org/10.1016/j.cam.2022.114780</a>","short":"S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics 421 (2023) 114780.","mla":"Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i>, vol. 421, Elsevier, 2023, p. 114780, doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>.","bibtex":"@article{Ober-Blöbaum_Offen_2023, title={Variational Learning of Euler–Lagrange Dynamics from Data}, volume={421}, DOI={<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier}, author={Ober-Blöbaum, Sina and Offen, Christian}, year={2023}, pages={114780} }","chicago":"Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i> 421 (2023): 114780. <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">https://doi.org/10.1016/j.cam.2022.114780</a>.","ieee":"S. Ober-Blöbaum and C. Offen, “Variational Learning of Euler–Lagrange Dynamics from Data,” <i>Journal of Computational and Applied Mathematics</i>, vol. 421, p. 114780, 2023, doi: <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>.","ama":"Ober-Blöbaum S, Offen C. Variational Learning of Euler–Lagrange Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>. 2023;421:114780. doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>"},"related_material":{"link":[{"url":"https://github.com/Christian-Offen/LagrangianShadowIntegration","relation":"software"}]},"publication_identifier":{"issn":["0377-0427"]},"has_accepted_license":"1","publication_status":"epub_ahead","doi":"10.1016/j.cam.2022.114780","volume":421,"author":[{"full_name":"Ober-Blöbaum, Sina","id":"16494","last_name":"Ober-Blöbaum","first_name":"Sina"},{"first_name":"Christian","id":"85279","full_name":"Offen, Christian","last_name":"Offen","orcid":"0000-0002-5940-8057"}],"date_updated":"2023-08-10T08:42:39Z","oa":"1","status":"public","type":"journal_article","file_date_updated":"2022-06-28T15:25:50Z","article_type":"original","department":[{"_id":"636"}],"user_id":"85279","_id":"29240"},{"file":[{"relation":"main_file","content_type":"application/pdf","description":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","file_size":827030,"title":"Backward error analysis for conjugate symplectic methods","file_id":"32801","access_level":"open_access","file_name":"BEA_MultiStep_Matrix.pdf","date_updated":"2022-08-12T16:48:59Z","date_created":"2022-08-12T16:48:59Z","creator":"coffen"}],"abstract":[{"text":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","lang":"eng"}],"publication":"Journal of Geometric Mechanics","language":[{"iso":"eng"}],"keyword":["variational integrators","backward error analysis","Euler--Lagrange equations","multistep methods","conjugate symplectic methods"],"ddc":["510"],"external_id":{"arxiv":["2201.03911"]},"year":"2023","issue":"1","quality_controlled":"1","title":"Backward error analysis for conjugate symplectic methods","date_created":"2022-01-11T12:48:39Z","publisher":"AIMS Press","status":"public","type":"journal_article","file_date_updated":"2022-08-12T16:48:59Z","article_type":"original","department":[{"_id":"636"}],"user_id":"85279","_id":"29236","page":"98-115","intvolume":"        15","citation":{"apa":"McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>","mla":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS Press, 2023, pp. 98–115, doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","bibtex":"@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate symplectic methods}, volume={15}, DOI={<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>}, number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press}, author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }","short":"R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115.","ama":"McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>","chicago":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>.","ieee":"R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023, doi: <a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>."},"related_material":{"link":[{"relation":"software","url":"https://github.com/Christian-Offen/BEAConjugateSymplectic"}]},"has_accepted_license":"1","publication_status":"published","doi":"10.3934/jgm.2023005","volume":15,"author":[{"full_name":"McLachlan, Robert","last_name":"McLachlan","first_name":"Robert"},{"first_name":"Christian","id":"85279","full_name":"Offen, Christian","orcid":"0000-0002-5940-8057","last_name":"Offen"}],"oa":"1","date_updated":"2023-08-10T08:40:30Z"}]