{"abstract":[{"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. ","lang":"eng"}],"related_material":{"link":[{"description":"GitHub","url":"https://github.com/Christian-Offen/DLNN_pde","relation":"software"}]},"title":"Learning of discrete models of variational PDEs from data","date_updated":"2024-01-09T11:29:06Z","file":[{"date_created":"2024-01-09T10:48:38Z","content_type":"application/pdf","file_id":"50376","access_level":"open_access","file_name":"Accepted manuscript with AIP banner CHA23-AR-01370.pdf","date_updated":"2024-01-09T10:48:38Z","title":"Accepted Manuscript Chaos","relation":"main_file","file_size":13222105,"creator":"coffen"},{"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.","creator":"coffen","date_created":"2024-01-09T11:19:49Z","access_level":"open_access","title":"Learning of discrete models of variational PDEs from data","relation":"main_file","file_size":12960884,"content_type":"application/pdf","file_id":"50390","file_name":"LDensityPDE_AIP.pdf","date_updated":"2024-01-09T11:19:49Z"}],"intvolume":"        34","issue":"1","department":[{"_id":"636"}],"has_accepted_license":"1","volume":34,"author":[{"id":"85279","first_name":"Christian","last_name":"Offen","full_name":"Offen, Christian","orcid":"0000-0002-5940-8057"},{"last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina","first_name":"Sina","id":"16494"}],"publication_identifier":{"issn":["1054-1500"]},"file_date_updated":"2024-01-09T11:19:49Z","doi":"10.1063/5.0172287","date_created":"2023-08-10T08:24:48Z","publication_status":"published","external_id":{"arxiv":["2308.05082 "]},"publication":"Chaos","quality_controlled":"1","language":[{"iso":"eng"}],"article_number":"013104","citation":{"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>","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>.","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>.","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>.","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} }","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>","short":"C. Offen, S. Ober-Blöbaum, Chaos 34 (2024)."},"ddc":["510"],"publisher":"AIP Publishing","type":"journal_article","oa":"1","year":"2024","user_id":"85279","_id":"46469","status":"public","article_type":"original"}