--- _id: '46469' 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. ' article_number: '013104' article_type: original author: - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum citation: ama: Offen C, Ober-Blöbaum S. Learning of discrete models of variational PDEs from data. Chaos. 2024;34(1). doi:10.1063/5.0172287 apa: Offen, C., & Ober-Blöbaum, S. (2024). Learning of discrete models of variational PDEs from data. Chaos, 34(1), Article 013104. https://doi.org/10.1063/5.0172287 bibtex: '@article{Offen_Ober-Blöbaum_2024, title={Learning of discrete models of variational PDEs from data}, volume={34}, DOI={10.1063/5.0172287}, number={1013104}, journal={Chaos}, publisher={AIP Publishing}, author={Offen, Christian and Ober-Blöbaum, Sina}, year={2024} }' chicago: Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” Chaos 34, no. 1 (2024). https://doi.org/10.1063/5.0172287. ieee: 'C. Offen and S. Ober-Blöbaum, “Learning of discrete models of variational PDEs from data,” Chaos, vol. 34, no. 1, Art. no. 013104, 2024, doi: 10.1063/5.0172287.' mla: Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” Chaos, vol. 34, no. 1, 013104, AIP Publishing, 2024, doi:10.1063/5.0172287. short: C. Offen, S. Ober-Blöbaum, Chaos 34 (2024). date_created: 2023-08-10T08:24:48Z date_updated: 2024-01-09T11:29:06Z ddc: - '510' department: - _id: '636' doi: 10.1063/5.0172287 external_id: arxiv: - '2308.05082 ' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2024-01-09T10:48:38Z date_updated: 2024-01-09T10:48:38Z file_id: '50376' file_name: Accepted manuscript with AIP banner CHA23-AR-01370.pdf file_size: 13222105 relation: main_file title: Accepted Manuscript Chaos - access_level: open_access content_type: application/pdf creator: coffen date_created: 2024-01-09T11:19:49Z date_updated: 2024-01-09T11:19:49Z description: |- 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. file_id: '50390' file_name: LDensityPDE_AIP.pdf file_size: 12960884 relation: main_file title: Learning of discrete models of variational PDEs from data file_date_updated: 2024-01-09T11:19:49Z has_accepted_license: '1' intvolume: ' 34' issue: '1' language: - iso: eng oa: '1' publication: Chaos publication_identifier: issn: - 1054-1500 publication_status: published publisher: AIP Publishing quality_controlled: '1' related_material: link: - description: GitHub relation: software url: https://github.com/Christian-Offen/DLNN_pde status: public title: Learning of discrete models of variational PDEs from data type: journal_article user_id: '85279' volume: 34 year: '2024' ... --- _id: '37654' abstract: - lang: eng 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." article_number: '063115' article_type: original author: - first_name: Eva full_name: Dierkes, Eva last_name: Dierkes - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Kathrin full_name: Flaßkamp, Kathrin last_name: Flaßkamp citation: ama: Dierkes E, Offen C, Ober-Blöbaum S, Flaßkamp K. Hamiltonian Neural Networks with Automatic Symmetry Detection. Chaos. 2023;33(6). doi:10.1063/5.0142969 apa: Dierkes, E., Offen, C., Ober-Blöbaum, S., & Flaßkamp, K. (2023). Hamiltonian Neural Networks with Automatic Symmetry Detection. Chaos, 33(6), Article 063115. https://doi.org/10.1063/5.0142969 bibtex: '@article{Dierkes_Offen_Ober-Blöbaum_Flaßkamp_2023, title={Hamiltonian Neural Networks with Automatic Symmetry Detection}, volume={33}, DOI={10.1063/5.0142969}, number={6063115}, journal={Chaos}, publisher={AIP Publishing}, author={Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}, year={2023} }' chicago: Dierkes, Eva, Christian Offen, Sina Ober-Blöbaum, and Kathrin Flaßkamp. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” Chaos 33, no. 6 (2023). https://doi.org/10.1063/5.0142969. ieee: 'E. Dierkes, C. Offen, S. Ober-Blöbaum, and K. Flaßkamp, “Hamiltonian Neural Networks with Automatic Symmetry Detection,” Chaos, vol. 33, no. 6, Art. no. 063115, 2023, doi: 10.1063/5.0142969.' mla: Dierkes, Eva, et al. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” Chaos, vol. 33, no. 6, 063115, AIP Publishing, 2023, doi:10.1063/5.0142969. short: E. Dierkes, C. Offen, S. Ober-Blöbaum, K. Flaßkamp, Chaos 33 (2023). date_created: 2023-01-20T09:10:06Z date_updated: 2023-08-10T08:37:01Z ddc: - '510' department: - _id: '636' doi: 10.1063/5.0142969 external_id: arxiv: - '2301.07928' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2023-04-26T16:20:56Z date_updated: 2023-04-26T16:20:56Z description: |- Incorporating physical system knowledge into data-driven system identification has been shown to be beneficial. The approach presented in this article combines learning of an energy-conserving model from data with detecting a Lie group representation of the unknown system symmetry. The proposed approach can improve the learned model and reveal underlying symmetry simultaneously. file_id: '44205' file_name: JournalPaper_main.pdf file_size: 5200111 relation: main_file title: Hamiltonian Neural Networks with Automatic Symmetry Detection file_date_updated: 2023-04-26T16:20:56Z has_accepted_license: '1' intvolume: ' 33' issue: '6' language: - iso: eng oa: '1' publication: Chaos publication_identifier: issn: - 1054-1500 publication_status: published publisher: AIP Publishing related_material: link: - description: GitHub relation: software url: https://github.com/eva-dierkes/HNN_withSymmetries status: public title: Hamiltonian Neural Networks with Automatic Symmetry Detection type: journal_article user_id: '85279' volume: 33 year: '2023' ... --- _id: '16552' author: - first_name: Michael full_name: Dellnitz, Michael last_name: Dellnitz - first_name: Andreas full_name: Hohmann, Andreas last_name: Hohmann - first_name: Oliver full_name: Junge, Oliver last_name: Junge - first_name: Martin full_name: Rumpf, Martin last_name: Rumpf citation: ama: 'Dellnitz M, Hohmann A, Junge O, Rumpf M. Exploring invariant sets and invariant measures. Chaos: An Interdisciplinary Journal of Nonlinear Science. 1997:221-228. doi:10.1063/1.166223' apa: 'Dellnitz, M., Hohmann, A., Junge, O., & Rumpf, M. (1997). Exploring invariant sets and invariant measures. Chaos: An Interdisciplinary Journal of Nonlinear Science, 221–228. https://doi.org/10.1063/1.166223' bibtex: '@article{Dellnitz_Hohmann_Junge_Rumpf_1997, title={Exploring invariant sets and invariant measures}, DOI={10.1063/1.166223}, 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} }' chicago: 'Dellnitz, Michael, Andreas Hohmann, Oliver Junge, and Martin Rumpf. “Exploring Invariant Sets and Invariant Measures.” Chaos: An Interdisciplinary Journal of Nonlinear Science, 1997, 221–28. https://doi.org/10.1063/1.166223.' ieee: 'M. Dellnitz, A. Hohmann, O. Junge, and M. Rumpf, “Exploring invariant sets and invariant measures,” Chaos: An Interdisciplinary Journal of Nonlinear Science, pp. 221–228, 1997.' mla: 'Dellnitz, Michael, et al. “Exploring Invariant Sets and Invariant Measures.” Chaos: An Interdisciplinary Journal of Nonlinear Science, 1997, pp. 221–28, doi:10.1063/1.166223.' short: 'M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Chaos: An Interdisciplinary Journal of Nonlinear Science (1997) 221–228.' date_created: 2020-04-15T09:06:28Z date_updated: 2022-01-06T06:52:52Z department: - _id: '101' doi: 10.1063/1.166223 language: - iso: eng page: 221-228 publication: 'Chaos: An Interdisciplinary Journal of Nonlinear Science' publication_identifier: issn: - 1054-1500 - 1089-7682 publication_status: published status: public title: Exploring invariant sets and invariant measures type: journal_article user_id: '15701' year: '1997' ...