[{"citation":{"ama":"Offen C, Ober-Blöbaum S. Learning of discrete models of variational PDEs from data. <i>Chaos</i>. 2024;34(1). doi:<a href=\"https://doi.org/10.1063/5.0172287\">10.1063/5.0172287</a>","chicago":"Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” <i>Chaos</i> 34, no. 1 (2024). <a href=\"https://doi.org/10.1063/5.0172287\">https://doi.org/10.1063/5.0172287</a>.","ieee":"C. Offen and S. Ober-Blöbaum, “Learning of discrete models of variational PDEs from data,” <i>Chaos</i>, vol. 34, no. 1, Art. no. 013104, 2024, doi: <a href=\"https://doi.org/10.1063/5.0172287\">10.1063/5.0172287</a>.","short":"C. Offen, S. Ober-Blöbaum, Chaos 34 (2024).","bibtex":"@article{Offen_Ober-Blöbaum_2024, title={Learning of discrete models of variational PDEs from data}, volume={34}, DOI={<a href=\"https://doi.org/10.1063/5.0172287\">10.1063/5.0172287</a>}, number={1013104}, journal={Chaos}, publisher={AIP Publishing}, author={Offen, Christian and Ober-Blöbaum, Sina}, year={2024} }","mla":"Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” <i>Chaos</i>, vol. 34, no. 1, 013104, AIP Publishing, 2024, doi:<a href=\"https://doi.org/10.1063/5.0172287\">10.1063/5.0172287</a>.","apa":"Offen, C., &#38; Ober-Blöbaum, S. (2024). Learning of discrete models of variational PDEs from data. <i>Chaos</i>, <i>34</i>(1), Article 013104. <a href=\"https://doi.org/10.1063/5.0172287\">https://doi.org/10.1063/5.0172287</a>"},"intvolume":"        34","related_material":{"link":[{"url":"https://github.com/Christian-Offen/DLNN_pde","description":"GitHub","relation":"software"}]},"publication_status":"published","has_accepted_license":"1","publication_identifier":{"issn":["1054-1500"]},"doi":"10.1063/5.0172287","author":[{"orcid":"0000-0002-5940-8057","last_name":"Offen","full_name":"Offen, Christian","id":"85279","first_name":"Christian"},{"first_name":"Sina","last_name":"Ober-Blöbaum","id":"16494","full_name":"Ober-Blöbaum, Sina"}],"volume":34,"oa":"1","date_updated":"2024-08-12T13:45:43Z","status":"public","type":"journal_article","file_date_updated":"2024-01-09T11:19:49Z","article_number":"013104","article_type":"original","user_id":"85279","department":[{"_id":"636"}],"project":[{"_id":"52","name":"PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing"}],"_id":"46469","year":"2024","issue":"1","quality_controlled":"1","title":"Learning of discrete models of variational PDEs from data","date_created":"2023-08-10T08:24:48Z","publisher":"AIP Publishing","file":[{"file_size":13222105,"title":"Accepted Manuscript Chaos","file_id":"50376","file_name":"Accepted manuscript with AIP banner CHA23-AR-01370.pdf","access_level":"open_access","date_updated":"2024-01-09T10:48:38Z","date_created":"2024-01-09T10:48:38Z","creator":"coffen","relation":"main_file","content_type":"application/pdf"},{"content_type":"application/pdf","relation":"main_file","date_updated":"2024-01-09T11:19:49Z","creator":"coffen","date_created":"2024-01-09T11:19:49Z","title":"Learning of discrete models of variational PDEs from data","file_size":12960884,"description":"We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train\na neural network model of a discrete Lagrangian density such that the discrete Euler–Lagrange equations are consistent\nwith the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian\ndensities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for\nthe training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers\nguarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical\nsimulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory\nsuch as travelling waves. This is possible even when travelling waves are not present in the training data. This is\ncompared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces\ncontaining structurally simple solutions when these are not present in the training data. Ideas are demonstrated on\nexamples based on the wave equation and the Schrödinger equation.","file_name":"LDensityPDE_AIP.pdf","file_id":"50390","access_level":"open_access"}],"abstract":[{"lang":"eng","text":"We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation. "}],"publication":"Chaos","language":[{"iso":"eng"}],"ddc":["510"],"external_id":{"arxiv":["2308.05082 "]}},{"status":"public","type":"journal_article","file_date_updated":"2023-04-26T16:20:56Z","article_type":"original","article_number":"063115","user_id":"85279","department":[{"_id":"636"}],"_id":"37654","citation":{"ama":"Dierkes E, Offen C, Ober-Blöbaum S, Flaßkamp K. Hamiltonian Neural Networks with Automatic Symmetry Detection. <i>Chaos</i>. 2023;33(6). doi:<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>","chicago":"Dierkes, Eva, Christian Offen, Sina Ober-Blöbaum, and Kathrin Flaßkamp. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” <i>Chaos</i> 33, no. 6 (2023). <a href=\"https://doi.org/10.1063/5.0142969\">https://doi.org/10.1063/5.0142969</a>.","ieee":"E. Dierkes, C. Offen, S. Ober-Blöbaum, and K. Flaßkamp, “Hamiltonian Neural Networks with Automatic Symmetry Detection,” <i>Chaos</i>, vol. 33, no. 6, Art. no. 063115, 2023, doi: <a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>.","short":"E. Dierkes, C. Offen, S. Ober-Blöbaum, K. Flaßkamp, Chaos 33 (2023).","bibtex":"@article{Dierkes_Offen_Ober-Blöbaum_Flaßkamp_2023, title={Hamiltonian Neural Networks with Automatic Symmetry Detection}, volume={33}, DOI={<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>}, number={6063115}, journal={Chaos}, publisher={AIP Publishing}, author={Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}, year={2023} }","mla":"Dierkes, Eva, et al. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” <i>Chaos</i>, vol. 33, no. 6, 063115, AIP Publishing, 2023, doi:<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>.","apa":"Dierkes, E., Offen, C., Ober-Blöbaum, S., &#38; Flaßkamp, K. (2023). Hamiltonian Neural Networks with Automatic Symmetry Detection. <i>Chaos</i>, <i>33</i>(6), Article 063115. <a href=\"https://doi.org/10.1063/5.0142969\">https://doi.org/10.1063/5.0142969</a>"},"intvolume":"        33","related_material":{"link":[{"url":"https://github.com/eva-dierkes/HNN_withSymmetries","relation":"software","description":"GitHub"}]},"publication_status":"published","publication_identifier":{"issn":["1054-1500"]},"has_accepted_license":"1","doi":"10.1063/5.0142969","author":[{"full_name":"Dierkes, Eva","last_name":"Dierkes","first_name":"Eva"},{"first_name":"Christian","last_name":"Offen","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian","id":"85279"},{"full_name":"Ober-Blöbaum, Sina","id":"16494","last_name":"Ober-Blöbaum","first_name":"Sina"},{"full_name":"Flaßkamp, Kathrin","last_name":"Flaßkamp","first_name":"Kathrin"}],"volume":33,"date_updated":"2023-08-10T08:37:01Z","oa":"1","file":[{"date_updated":"2023-04-26T16:20:56Z","creator":"coffen","date_created":"2023-04-26T16:20:56Z","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","file_size":5200111,"description":"Incorporating physical system knowledge into data-driven\nsystem identification has been shown to be beneficial. The\napproach presented in this article combines learning of an\nenergy-conserving model from data with detecting a Lie\ngroup representation of the unknown system symmetry.\nThe proposed approach can improve the learned model\nand reveal underlying symmetry simultaneously.","file_name":"JournalPaper_main.pdf","access_level":"open_access","file_id":"44205","content_type":"application/pdf","relation":"main_file"}],"abstract":[{"text":"Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when\r\nlearning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite\r\nthe data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we\r\nenhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach\r\nallows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples,\r\na pendulum on a cart and a two-body problem from astrodynamics are considered.","lang":"eng"}],"publication":"Chaos","language":[{"iso":"eng"}],"ddc":["510"],"external_id":{"arxiv":["2301.07928"]},"year":"2023","issue":"6","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","date_created":"2023-01-20T09:10:06Z","publisher":"AIP Publishing"},{"_id":"16552","user_id":"15701","department":[{"_id":"101"}],"language":[{"iso":"eng"}],"type":"journal_article","publication":"Chaos: An Interdisciplinary Journal of Nonlinear Science","status":"public","date_updated":"2022-01-06T06:52:52Z","date_created":"2020-04-15T09:06:28Z","author":[{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"},{"last_name":"Hohmann","full_name":"Hohmann, Andreas","first_name":"Andreas"},{"last_name":"Junge","full_name":"Junge, Oliver","first_name":"Oliver"},{"first_name":"Martin","full_name":"Rumpf, Martin","last_name":"Rumpf"}],"title":"Exploring invariant sets and invariant measures","doi":"10.1063/1.166223","publication_status":"published","publication_identifier":{"issn":["1054-1500","1089-7682"]},"year":"1997","citation":{"bibtex":"@article{Dellnitz_Hohmann_Junge_Rumpf_1997, title={Exploring invariant sets and invariant measures}, DOI={<a href=\"https://doi.org/10.1063/1.166223\">10.1063/1.166223</a>}, journal={Chaos: An Interdisciplinary Journal of Nonlinear Science}, author={Dellnitz, Michael and Hohmann, Andreas and Junge, Oliver and Rumpf, Martin}, year={1997}, pages={221–228} }","short":"M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Chaos: An Interdisciplinary Journal of Nonlinear Science (1997) 221–228.","mla":"Dellnitz, Michael, et al. “Exploring Invariant Sets and Invariant Measures.” <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>, 1997, pp. 221–28, doi:<a href=\"https://doi.org/10.1063/1.166223\">10.1063/1.166223</a>.","apa":"Dellnitz, M., Hohmann, A., Junge, O., &#38; Rumpf, M. (1997). Exploring invariant sets and invariant measures. <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>, 221–228. <a href=\"https://doi.org/10.1063/1.166223\">https://doi.org/10.1063/1.166223</a>","ieee":"M. Dellnitz, A. Hohmann, O. Junge, and M. Rumpf, “Exploring invariant sets and invariant measures,” <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>, pp. 221–228, 1997.","chicago":"Dellnitz, Michael, Andreas Hohmann, Oliver Junge, and Martin Rumpf. “Exploring Invariant Sets and Invariant Measures.” <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>, 1997, 221–28. <a href=\"https://doi.org/10.1063/1.166223\">https://doi.org/10.1063/1.166223</a>.","ama":"Dellnitz M, Hohmann A, Junge O, Rumpf M. Exploring invariant sets and invariant measures. <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>. 1997:221-228. doi:<a href=\"https://doi.org/10.1063/1.166223\">10.1063/1.166223</a>"},"page":"221-228"}]