--- _id: '59794' abstract: - lang: eng text: The depth of networks plays a crucial role in the effectiveness of deep learning. However, the memory requirement for backpropagation scales linearly with the number of layers, which leads to memory bottlenecks during training. Moreover, deep networks are often unable to handle time-series data appearing at irregular intervals. These issues can be resolved by considering continuous-depth networks based on the neural ODE framework in combination with reversible integration methods that allow for variable time-steps. Reversibility of the method ensures that the memory requirement for training is independent of network depth, while variable time-steps are required for assimilating time-series data on irregular intervals. However, at present, there are no known higher-order reversible methods with this property. High-order methods are especially important when a high level of accuracy in learning is required or when small time-steps are necessary due to large errors in time integration of neural ODEs, for instance in context of complex dynamical systems such as Kepler systems and molecular dynamics. The requirement of small time-steps when using a low-order method can significantly increase the computational cost of training as well as inference. In this work, we present an approach for constructing high-order reversible methods that allow adaptive time-stepping. Our numerical tests show the advantages in computational speed when applied to the task of learning dynamical systems. author: - first_name: Sofya full_name: Maslovskaya, Sofya id: '87909' last_name: Maslovskaya - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 - first_name: Pranav full_name: Singh, Pranav last_name: Singh - first_name: Boris Edgar full_name: Wembe Moafo, Boris Edgar id: '95394' last_name: Wembe Moafo citation: ama: Maslovskaya S, Ober-Blöbaum S, Offen C, Singh P, Wembe Moafo BE. Adaptive higher order reversible integrators for memory efficient deep learning. Published online 2025. apa: Maslovskaya, S., Ober-Blöbaum, S., Offen, C., Singh, P., & Wembe Moafo, B. E. (2025). Adaptive higher order reversible integrators for memory efficient deep learning. bibtex: '@article{Maslovskaya_Ober-Blöbaum_Offen_Singh_Wembe Moafo_2025, title={Adaptive higher order reversible integrators for memory efficient deep learning}, author={Maslovskaya, Sofya and Ober-Blöbaum, Sina and Offen, Christian and Singh, Pranav and Wembe Moafo, Boris Edgar}, year={2025} }' chicago: Maslovskaya, Sofya, Sina Ober-Blöbaum, Christian Offen, Pranav Singh, and Boris Edgar Wembe Moafo. “Adaptive Higher Order Reversible Integrators for Memory Efficient Deep Learning,” 2025. ieee: S. Maslovskaya, S. Ober-Blöbaum, C. Offen, P. Singh, and B. E. Wembe Moafo, “Adaptive higher order reversible integrators for memory efficient deep learning.” 2025. mla: Maslovskaya, Sofya, et al. Adaptive Higher Order Reversible Integrators for Memory Efficient Deep Learning. 2025. short: S. Maslovskaya, S. Ober-Blöbaum, C. Offen, P. Singh, B.E. Wembe Moafo, (2025). date_created: 2025-05-05T09:25:28Z date_updated: 2025-09-30T15:16:09Z ddc: - '510' department: - _id: '636' external_id: arxiv: - '2410.09537' file: - access_level: closed content_type: application/pdf creator: sofyam date_created: 2025-05-05T09:28:02Z date_updated: 2025-05-05T09:28:02Z file_id: '59795' file_name: 2410.09537v2.pdf file_size: 1830758 relation: main_file success: 1 file_date_updated: 2025-05-05T09:28:02Z has_accepted_license: '1' language: - iso: eng status: public title: Adaptive higher order reversible integrators for memory efficient deep learning type: preprint user_id: '85279' year: '2025' ... --- _id: '62980' abstract: - lang: eng text: We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which is in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes. article_number: '033062' author: - first_name: Denis A. full_name: Kopylov, Denis A. last_name: Kopylov - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 - first_name: Laura full_name: Ares, Laura last_name: Ares - first_name: Boris Edgar full_name: Wembe Moafo, Boris Edgar id: '95394' last_name: Wembe Moafo - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Torsten full_name: Meier, Torsten id: '344' last_name: Meier orcid: 0000-0001-8864-2072 - first_name: Polina R. full_name: Sharapova, Polina R. id: '60286' last_name: Sharapova - first_name: Jan full_name: Sperling, Jan id: '75127' last_name: Sperling orcid: 0000-0002-5844-3205 citation: ama: Kopylov DA, Offen C, Ares L, et al. Multiphoton, multimode state classification for nonlinear optical circuits. Physical Review Research. 2025;7(3). doi:10.1103/sv6z-v1gk apa: Kopylov, D. A., Offen, C., Ares, L., Wembe Moafo, B. E., Ober-Blöbaum, S., Meier, T., Sharapova, P. R., & Sperling, J. (2025). Multiphoton, multimode state classification for nonlinear optical circuits. Physical Review Research, 7(3), Article 033062. https://doi.org/10.1103/sv6z-v1gk bibtex: '@article{Kopylov_Offen_Ares_Wembe Moafo_Ober-Blöbaum_Meier_Sharapova_Sperling_2025, title={Multiphoton, multimode state classification for nonlinear optical circuits}, volume={7}, DOI={10.1103/sv6z-v1gk}, number={3033062}, journal={Physical Review Research}, publisher={American Physical Society (APS)}, author={Kopylov, Denis A. and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina R. and Sperling, Jan}, year={2025} }' chicago: Kopylov, Denis A., Christian Offen, Laura Ares, Boris Edgar Wembe Moafo, Sina Ober-Blöbaum, Torsten Meier, Polina R. Sharapova, and Jan Sperling. “Multiphoton, Multimode State Classification for Nonlinear Optical Circuits.” Physical Review Research 7, no. 3 (2025). https://doi.org/10.1103/sv6z-v1gk. ieee: 'D. A. Kopylov et al., “Multiphoton, multimode state classification for nonlinear optical circuits,” Physical Review Research, vol. 7, no. 3, Art. no. 033062, 2025, doi: 10.1103/sv6z-v1gk.' mla: Kopylov, Denis A., et al. “Multiphoton, Multimode State Classification for Nonlinear Optical Circuits.” Physical Review Research, vol. 7, no. 3, 033062, American Physical Society (APS), 2025, doi:10.1103/sv6z-v1gk. short: D.A. Kopylov, C. Offen, L. Ares, B.E. Wembe Moafo, S. Ober-Blöbaum, T. Meier, P.R. Sharapova, J. Sperling, Physical Review Research 7 (2025). date_created: 2025-12-09T09:08:39Z date_updated: 2025-12-09T09:10:01Z department: - _id: '15' - _id: '569' - _id: '170' - _id: '293' - _id: '706' - _id: '636' - _id: '35' - _id: '230' - _id: '429' - _id: '623' doi: 10.1103/sv6z-v1gk intvolume: ' 7' issue: '3' language: - iso: eng project: - _id: '53' name: 'TRR 142: Maßgeschneiderte nichtlineare Photonik: Von grundlegenden Konzepten zu funktionellen Strukturen' - _id: '56' name: TRR 142 - Project Area C - _id: '174' name: 'TRR 142 ; TP: C10: Erzeugung und Charakterisierung von Quantenlicht in nichtlinearen Systemen: Eine theoretische Analyse' - _id: '266' name: 'PhoQC: Photonisches Quantencomputing' publication: Physical Review Research publication_identifier: issn: - 2643-1564 publication_status: published publisher: American Physical Society (APS) status: public title: Multiphoton, multimode state classification for nonlinear optical circuits type: journal_article user_id: '16199' volume: 7 year: '2025' ... --- _id: '58544' abstract: - lang: eng text: 'We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes. ' author: - first_name: Denis full_name: Kopylov, Denis id: '98502' last_name: Kopylov - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 - first_name: Laura full_name: Ares, Laura last_name: Ares - first_name: Boris Edgar full_name: Wembe Moafo, Boris Edgar id: '95394' last_name: Wembe Moafo - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Torsten full_name: Meier, Torsten id: '344' last_name: Meier orcid: 0000-0001-8864-2072 - first_name: Polina full_name: Sharapova, Polina id: '60286' last_name: Sharapova - first_name: Jan full_name: Sperling, Jan id: '75127' last_name: Sperling orcid: 0000-0002-5844-3205 citation: ama: Kopylov D, Offen C, Ares L, et al. Multiphoton, multimode state classification for nonlinear optical circuits . apa: Kopylov, D., Offen, C., Ares, L., Wembe Moafo, B. E., Ober-Blöbaum, S., Meier, T., Sharapova, P., & Sperling, J. (n.d.). Multiphoton, multimode state classification for nonlinear optical circuits . bibtex: '@article{Kopylov_Offen_Ares_Wembe Moafo_Ober-Blöbaum_Meier_Sharapova_Sperling, title={Multiphoton, multimode state classification for nonlinear optical circuits }, author={Kopylov, Denis and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina and Sperling, Jan} }' chicago: Kopylov, Denis, Christian Offen, Laura Ares, Boris Edgar Wembe Moafo, Sina Ober-Blöbaum, Torsten Meier, Polina Sharapova, and Jan Sperling. “Multiphoton, Multimode State Classification for Nonlinear Optical Circuits ,” n.d. ieee: D. Kopylov et al., “Multiphoton, multimode state classification for nonlinear optical circuits .” . mla: Kopylov, Denis, et al. Multiphoton, Multimode State Classification for Nonlinear Optical Circuits . short: D. Kopylov, C. Offen, L. Ares, B.E. Wembe Moafo, S. Ober-Blöbaum, T. Meier, P. Sharapova, J. Sperling, (n.d.). date_created: 2025-02-10T08:26:45Z date_updated: 2025-02-10T08:36:12Z department: - _id: '623' - _id: '15' - _id: '636' external_id: arxiv: - '2502.05123' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2502.05123 oa: '1' publication_status: submitted status: public title: 'Multiphoton, multimode state classification for nonlinear optical circuits ' type: preprint user_id: '85279' year: '2025' ... --- _id: '53805' abstract: - lang: eng text: The article introduces a method to learn dynamical systems that are governed by Euler–Lagrange 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. article_type: original author: - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 citation: ama: Offen C. Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification. Mathematics of Computation. Published online 2025. doi:10.1090/mcom/4120 apa: Offen, C. (2025). Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification. Mathematics of Computation. https://doi.org/10.1090/mcom/4120 bibtex: '@article{Offen_2025, title={Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification}, DOI={10.1090/mcom/4120}, journal={Mathematics of Computation}, publisher={American Mathematical Society}, author={Offen, Christian}, year={2025} }' chicago: Offen, Christian. “Machine Learning of Continuous and Discrete Variational ODEs with Convergence Guarantee and Uncertainty Quantification.” Mathematics of Computation, 2025. https://doi.org/10.1090/mcom/4120. ieee: 'C. Offen, “Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification,” Mathematics of Computation, 2025, doi: 10.1090/mcom/4120.' mla: Offen, Christian. “Machine Learning of Continuous and Discrete Variational ODEs with Convergence Guarantee and Uncertainty Quantification.” Mathematics of Computation, American Mathematical Society, 2025, doi:10.1090/mcom/4120. short: C. Offen, Mathematics of Computation (2025). date_created: 2024-04-30T16:04:40Z date_updated: 2025-06-29T13:03:55Z ddc: - '510' department: - _id: '636' doi: 10.1090/mcom/4120 external_id: arxiv: - arXiv:2404.19626 file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2025-05-02T13:20:31Z date_updated: 2025-05-02T13:20:31Z description: |- The article introduces a method to learn dynamical systems that are governed by Euler–Lagrange 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 con- vergence 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. file_id: '59759' file_name: L_Collocation_ODE_mcom-l-template.pdf file_size: 1819189 relation: main_file title: 'Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification ' file_date_updated: 2025-05-02T13:20:31Z has_accepted_license: '1' language: - iso: eng oa: '1' project: - _id: '52' name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing' publication: Mathematics of Computation publication_status: epub_ahead publisher: American Mathematical Society quality_controlled: '1' related_material: link: - description: GitHub relation: software url: https://github.com/Christian-Offen/Lagrangian_GP status: public title: Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification type: journal_article user_id: '85279' year: '2025' ... --- _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: 2024-08-12T13:45:43Z 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' project: - _id: '52' name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing' 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: '55159' abstract: - lang: eng text: "We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus.\r\nMoreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schrödinger equation.\r\nThe article constitutes an extension of our previous article arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes." author: - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 citation: ama: Offen C. Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification. apa: Offen, C. (n.d.). Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification. bibtex: '@article{Offen, title={Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification}, author={Offen, Christian} }' chicago: Offen, Christian. “Machine Learning of Discrete Field Theories with Guaranteed Convergence and Uncertainty Quantification,” n.d. ieee: C. Offen, “Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification.” . mla: Offen, Christian. Machine Learning of Discrete Field Theories with Guaranteed Convergence and Uncertainty Quantification. short: C. Offen, (n.d.). date_created: 2024-07-10T13:43:50Z date_updated: 2024-08-12T13:43:32Z ddc: - '510' department: - _id: '636' external_id: arxiv: - '2407.07642' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2024-07-10T13:39:32Z date_updated: 2024-07-10T13:39:32Z description: |- We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schrödinger equation. The article constitutes an extension of our previous article for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes. file_id: '55160' file_name: L_Collocation.pdf file_size: 4569314 relation: main_file title: Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification file_date_updated: 2024-07-10T13:39:32Z has_accepted_license: '1' keyword: - System identification - inverse problem of variational calculus - Gaussian process - Lagrangian learning - physics informed machine learning - geometry aware learning language: - iso: eng oa: '1' page: '28' project: - _id: '52' name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing' publication_status: submitted related_material: link: - description: GitHub relation: software url: https://github.com/Christian-Offen/Lagrangian_GP_PDE status: public title: Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification type: preprint user_id: '85279' year: '2024' ... --- _id: '34135' abstract: - lang: eng text: By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function Ld which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise. author: - first_name: Yana full_name: Lishkova, Yana last_name: Lishkova - first_name: Paul full_name: Scherer, Paul last_name: Scherer - first_name: Steffen full_name: Ridderbusch, Steffen last_name: Ridderbusch - first_name: Mateja full_name: Jamnik, Mateja last_name: Jamnik - first_name: Pietro full_name: Liò, Pietro last_name: Liò - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 citation: ama: 'Lishkova Y, Scherer P, Ridderbusch S, et al. Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery. In: IFAC-PapersOnLine. Vol 56. Elsevier; 2023:3203-3210. doi:10.1016/j.ifacol.2023.10.1457' apa: Lishkova, Y., Scherer, P., Ridderbusch, S., Jamnik, M., Liò, P., Ober-Blöbaum, S., & Offen, C. (2023). Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery. IFAC-PapersOnLine, 56(2), 3203–3210. https://doi.org/10.1016/j.ifacol.2023.10.1457 bibtex: '@inproceedings{Lishkova_Scherer_Ridderbusch_Jamnik_Liò_Ober-Blöbaum_Offen_2023, title={Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery}, volume={56}, DOI={10.1016/j.ifacol.2023.10.1457}, number={2}, booktitle={IFAC-PapersOnLine}, publisher={Elsevier}, author={Lishkova, Yana and Scherer, Paul and Ridderbusch, Steffen and Jamnik, Mateja and Liò, Pietro and Ober-Blöbaum, Sina and Offen, Christian}, year={2023}, pages={3203–3210} }' chicago: Lishkova, Yana, Paul Scherer, Steffen Ridderbusch, Mateja Jamnik, Pietro Liò, Sina Ober-Blöbaum, and Christian Offen. “Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery.” In IFAC-PapersOnLine, 56:3203–10. Elsevier, 2023. https://doi.org/10.1016/j.ifacol.2023.10.1457. ieee: 'Y. Lishkova et al., “Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery,” in IFAC-PapersOnLine, Yokohama, Japan, 2023, vol. 56, no. 2, pp. 3203–3210, doi: 10.1016/j.ifacol.2023.10.1457.' mla: Lishkova, Yana, et al. “Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery.” IFAC-PapersOnLine, vol. 56, no. 2, Elsevier, 2023, pp. 3203–10, doi:10.1016/j.ifacol.2023.10.1457. short: 'Y. Lishkova, P. Scherer, S. Ridderbusch, M. Jamnik, P. Liò, S. Ober-Blöbaum, C. Offen, in: IFAC-PapersOnLine, Elsevier, 2023, pp. 3203–3210.' conference: end_date: 2023-07-14 location: ' Yokohama, Japan' name: The 22nd World Congress of the International Federation of Automatic Control start_date: 2023-07-09 date_created: 2022-11-23T08:17:10Z date_updated: 2023-12-29T14:26:00Z ddc: - '510' department: - _id: '636' doi: 10.1016/j.ifacol.2023.10.1457 external_id: arxiv: - '2211.10830' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2023-04-17T08:05:55Z date_updated: 2023-04-17T08:05:55Z description: |- By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function Ld which defines them. Based on ideas from Lie group theory, we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise. file_id: '44037' file_name: LNN_project.pdf file_size: 576115 relation: main_file title: Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery file_date_updated: 2023-04-17T08:05:55Z has_accepted_license: '1' intvolume: ' 56' issue: '2' language: - iso: eng main_file_link: - url: https://www.sciencedirect.com/science/article/pii/S2405896323018657 oa: '1' page: 3203-3210 publication: IFAC-PapersOnLine publication_status: published publisher: Elsevier quality_controlled: '1' related_material: link: - description: GitHub relation: software url: https://github.com/yanalish/SymDLNN status: public title: Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery type: conference user_id: '85279' volume: 56 year: '2023' ... --- _id: '42163' abstract: - lang: eng text: 'The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.' 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 discrete Lagrangians for variational PDEs from data and detection of travelling waves. In: Nielsen F, Barbaresco F, eds. Geometric Science of Information. Vol 14071. Lecture Notes in Computer Science (LNCS). Springer, Cham.; 2023:569-579. doi:10.1007/978-3-031-38271-0_57' apa: Offen, C., & Ober-Blöbaum, S. (2023). Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves. In F. Nielsen & F. Barbaresco (Eds.), Geometric Science of Information (Vol. 14071, pp. 569–579). Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_57 bibtex: '@inproceedings{Offen_Ober-Blöbaum_2023, series={Lecture Notes in Computer Science (LNCS)}, title={Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves}, volume={14071}, DOI={10.1007/978-3-031-38271-0_57}, booktitle={Geometric Science of Information}, publisher={Springer, Cham.}, author={Offen, Christian and Ober-Blöbaum, Sina}, editor={Nielsen, F and Barbaresco, F}, year={2023}, pages={569–579}, collection={Lecture Notes in Computer Science (LNCS)} }' chicago: Offen, Christian, and Sina Ober-Blöbaum. “Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves.” In Geometric Science of Information, edited by F Nielsen and F Barbaresco, 14071:569–79. Lecture Notes in Computer Science (LNCS). Springer, Cham., 2023. https://doi.org/10.1007/978-3-031-38271-0_57. ieee: 'C. Offen and S. Ober-Blöbaum, “Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves,” in Geometric Science of Information, Saint-Malo, Palais du Grand Large, France, 2023, vol. 14071, pp. 569–579, doi: 10.1007/978-3-031-38271-0_57.' mla: Offen, Christian, and Sina Ober-Blöbaum. “Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves.” Geometric Science of Information, edited by F Nielsen and F Barbaresco, vol. 14071, Springer, Cham., 2023, pp. 569–79, doi:10.1007/978-3-031-38271-0_57. short: 'C. Offen, S. Ober-Blöbaum, in: F. Nielsen, F. Barbaresco (Eds.), Geometric Science of Information, Springer, Cham., 2023, pp. 569–579.' conference: end_date: 2023-09-01 location: Saint-Malo, Palais du Grand Large, France name: ' GSI''23 6th International Conference on Geometric Science of Information' start_date: 2023-08-30 date_created: 2023-02-16T11:32:48Z date_updated: 2024-08-12T13:46:29Z ddc: - '510' department: - _id: '636' doi: 10.1007/978-3-031-38271-0_57 editor: - first_name: F full_name: Nielsen, F last_name: Nielsen - first_name: F full_name: Barbaresco, F last_name: Barbaresco external_id: arxiv: - '2302.08232 ' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2023-08-02T12:04:17Z date_updated: 2023-08-02T12:04:17Z description: |- The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler–Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE. file_id: '46273' file_name: LDensityLearning.pdf file_size: 1938962 relation: main_file title: Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves file_date_updated: 2023-08-02T12:04:17Z has_accepted_license: '1' intvolume: ' 14071' keyword: - System identification - discrete Lagrangians - travelling waves language: - iso: eng oa: '1' page: 569-579 project: - _id: '52' name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing' publication: Geometric Science of Information publication_identifier: eisbn: - 978-3-031-38271-0 publication_status: published publisher: Springer, Cham. quality_controlled: '1' related_material: link: - description: GitHub relation: software url: https://github.com/Christian-Offen/LagrangianDensityML series_title: Lecture Notes in Computer Science (LNCS) status: public title: Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves type: conference user_id: '85279' volume: 14071 year: '2023' ... --- _id: '29240' abstract: - lang: eng 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." article_type: original author: - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 citation: ama: Ober-Blöbaum S, Offen C. Variational Learning of Euler–Lagrange Dynamics from Data. Journal of Computational and Applied Mathematics. 2023;421:114780. doi:10.1016/j.cam.2022.114780 apa: Ober-Blöbaum, S., & Offen, C. (2023). Variational Learning of Euler–Lagrange Dynamics from Data. Journal of Computational and Applied Mathematics, 421, 114780. https://doi.org/10.1016/j.cam.2022.114780 bibtex: '@article{Ober-Blöbaum_Offen_2023, title={Variational Learning of Euler–Lagrange Dynamics from Data}, volume={421}, DOI={10.1016/j.cam.2022.114780}, 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.” Journal of Computational and Applied Mathematics 421 (2023): 114780. https://doi.org/10.1016/j.cam.2022.114780.' ieee: 'S. Ober-Blöbaum and C. Offen, “Variational Learning of Euler–Lagrange Dynamics from Data,” Journal of Computational and Applied Mathematics, vol. 421, p. 114780, 2023, doi: 10.1016/j.cam.2022.114780.' mla: Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” Journal of Computational and Applied Mathematics, vol. 421, Elsevier, 2023, p. 114780, doi:10.1016/j.cam.2022.114780. short: S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics 421 (2023) 114780. date_created: 2022-01-11T13:24:00Z date_updated: 2023-08-10T08:42:39Z ddc: - '510' department: - _id: '636' doi: 10.1016/j.cam.2022.114780 external_id: arxiv: - '2112.12619' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2022-06-28T15:25:50Z date_updated: 2022-06-28T15:25:50Z description: |- 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 equa- tions, 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 predic- tions 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, we 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. file_id: '32274' file_name: ShadowLagrangian_revision1_journal_style_arxiv.pdf file_size: 3640770 relation: main_file title: Variational Learning of Euler–Lagrange Dynamics from Data file_date_updated: 2022-06-28T15:25:50Z has_accepted_license: '1' intvolume: ' 421' keyword: - Lagrangian learning - variational backward error analysis - modified Lagrangian - variational integrators - physics informed learning language: - iso: eng oa: '1' page: '114780' publication: Journal of Computational and Applied Mathematics publication_identifier: issn: - 0377-0427 publication_status: epub_ahead publisher: Elsevier quality_controlled: '1' related_material: link: - relation: software url: https://github.com/Christian-Offen/LagrangianShadowIntegration status: public title: Variational Learning of Euler–Lagrange Dynamics from Data type: journal_article user_id: '85279' volume: 421 year: '2023' ... --- _id: '29236' abstract: - lang: eng 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. article_type: original author: - first_name: Robert full_name: McLachlan, Robert last_name: McLachlan - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 citation: ama: McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods. Journal of Geometric Mechanics. 2023;15(1):98-115. doi:10.3934/jgm.2023005 apa: McLachlan, R., & Offen, C. (2023). Backward error analysis for conjugate symplectic methods. Journal of Geometric Mechanics, 15(1), 98–115. https://doi.org/10.3934/jgm.2023005 bibtex: '@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate symplectic methods}, volume={15}, DOI={10.3934/jgm.2023005}, number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press}, author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }' chicago: 'McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” Journal of Geometric Mechanics 15, no. 1 (2023): 98–115. https://doi.org/10.3934/jgm.2023005.' ieee: 'R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic methods,” Journal of Geometric Mechanics, vol. 15, no. 1, pp. 98–115, 2023, doi: 10.3934/jgm.2023005.' mla: McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” Journal of Geometric Mechanics, vol. 15, no. 1, AIMS Press, 2023, pp. 98–115, doi:10.3934/jgm.2023005. short: R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115. date_created: 2022-01-11T12:48:39Z date_updated: 2023-08-10T08:40:30Z ddc: - '510' department: - _id: '636' doi: 10.3934/jgm.2023005 external_id: arxiv: - '2201.03911' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2022-08-12T16:48:59Z date_updated: 2022-08-12T16:48:59Z 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_id: '32801' file_name: BEA_MultiStep_Matrix.pdf file_size: 827030 relation: main_file title: Backward error analysis for conjugate symplectic methods file_date_updated: 2022-08-12T16:48:59Z has_accepted_license: '1' intvolume: ' 15' issue: '1' keyword: - variational integrators - backward error analysis - Euler--Lagrange equations - multistep methods - conjugate symplectic methods language: - iso: eng oa: '1' page: 98-115 publication: Journal of Geometric Mechanics publication_status: published publisher: AIMS Press quality_controlled: '1' related_material: link: - relation: software url: https://github.com/Christian-Offen/BEAConjugateSymplectic status: public title: Backward error analysis for conjugate symplectic methods type: journal_article user_id: '85279' volume: 15 year: '2023' ... --- _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: '21600' abstract: - lang: eng text: Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML. author: - first_name: Michael full_name: Dellnitz, Michael last_name: Dellnitz - first_name: Eyke full_name: Hüllermeier, Eyke id: '48129' last_name: Hüllermeier - first_name: Marvin full_name: Lücke, Marvin last_name: Lücke - first_name: Sina full_name: Ober-Blöbaum, Sina id: '16494' last_name: Ober-Blöbaum - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: 0000-0002-5940-8057 - first_name: Sebastian full_name: Peitz, Sebastian id: '47427' last_name: Peitz orcid: 0000-0002-3389-793X - first_name: Karlson full_name: Pfannschmidt, Karlson id: '13472' last_name: Pfannschmidt orcid: 0000-0001-9407-7903 citation: ama: Dellnitz M, Hüllermeier E, Lücke M, et al. Efficient time stepping for numerical integration using reinforcement  learning. SIAM Journal on Scientific Computing. 2023;45(2):A579-A595. doi:10.1137/21M1412682 apa: Dellnitz, M., Hüllermeier, E., Lücke, M., Ober-Blöbaum, S., Offen, C., Peitz, S., & Pfannschmidt, K. (2023). Efficient time stepping for numerical integration using reinforcement  learning. SIAM Journal on Scientific Computing, 45(2), A579–A595. https://doi.org/10.1137/21M1412682 bibtex: '@article{Dellnitz_Hüllermeier_Lücke_Ober-Blöbaum_Offen_Peitz_Pfannschmidt_2023, title={Efficient time stepping for numerical integration using reinforcement  learning}, volume={45}, DOI={10.1137/21M1412682}, number={2}, journal={SIAM Journal on Scientific Computing}, author={Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}, year={2023}, pages={A579–A595} }' chicago: 'Dellnitz, Michael, Eyke Hüllermeier, Marvin Lücke, Sina Ober-Blöbaum, Christian Offen, Sebastian Peitz, and Karlson Pfannschmidt. “Efficient Time Stepping for Numerical Integration Using Reinforcement  Learning.” SIAM Journal on Scientific Computing 45, no. 2 (2023): A579–95. https://doi.org/10.1137/21M1412682.' ieee: 'M. Dellnitz et al., “Efficient time stepping for numerical integration using reinforcement  learning,” SIAM Journal on Scientific Computing, vol. 45, no. 2, pp. A579–A595, 2023, doi: 10.1137/21M1412682.' mla: Dellnitz, Michael, et al. “Efficient Time Stepping for Numerical Integration Using Reinforcement  Learning.” SIAM Journal on Scientific Computing, vol. 45, no. 2, 2023, pp. A579–95, doi:10.1137/21M1412682. short: M. Dellnitz, E. Hüllermeier, M. Lücke, S. Ober-Blöbaum, C. Offen, S. Peitz, K. Pfannschmidt, SIAM Journal on Scientific Computing 45 (2023) A579–A595. date_created: 2021-04-09T07:59:19Z date_updated: 2023-08-25T09:24:50Z ddc: - '510' department: - _id: '101' - _id: '636' - _id: '355' - _id: '655' doi: 10.1137/21M1412682 external_id: arxiv: - arXiv:2104.03562 has_accepted_license: '1' intvolume: ' 45' issue: '2' language: - iso: eng main_file_link: - url: https://epubs.siam.org/doi/reader/10.1137/21M1412682 page: A579-A595 publication: SIAM Journal on Scientific Computing publication_status: published related_material: link: - description: GitHub relation: software url: https://github.com/lueckem/quadrature-ML status: public title: Efficient time stepping for numerical integration using reinforcement learning type: journal_article user_id: '47427' volume: 45 year: '2023' ... --- _id: '19941' abstract: - lang: eng text: "In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized\r\npartial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation." article_type: original author: - first_name: Robert I full_name: McLachlan, Robert I last_name: McLachlan - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: https://orcid.org/0000-0002-5940-8057 citation: ama: McLachlan RI, Offen C. Backward error analysis for variational discretisations of partial  differential equations. Journal of Geometric Mechanics. 2022;14(3):447-471. doi:10.3934/jgm.2022014 apa: McLachlan, R. I., & Offen, C. (2022). Backward error analysis for variational discretisations of partial  differential equations. Journal of Geometric Mechanics, 14(3), 447–471. https://doi.org/10.3934/jgm.2022014 bibtex: '@article{McLachlan_Offen_2022, title={Backward error analysis for variational discretisations of partial  differential equations}, volume={14}, DOI={10.3934/jgm.2022014}, number={3}, journal={Journal of Geometric Mechanics}, publisher={AIMS}, author={McLachlan, Robert I and Offen, Christian}, year={2022}, pages={447–471} }' chicago: 'McLachlan, Robert I, and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial  Differential Equations.” Journal of Geometric Mechanics 14, no. 3 (2022): 447–71. https://doi.org/10.3934/jgm.2022014.' ieee: 'R. I. McLachlan and C. Offen, “Backward error analysis for variational discretisations of partial  differential equations,” Journal of Geometric Mechanics, vol. 14, no. 3, pp. 447–471, 2022, doi: 10.3934/jgm.2022014.' mla: McLachlan, Robert I., and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial  Differential Equations.” Journal of Geometric Mechanics, vol. 14, no. 3, AIMS, 2022, pp. 447–71, doi:10.3934/jgm.2022014. short: R.I. McLachlan, C. Offen, Journal of Geometric Mechanics 14 (2022) 447–471. date_created: 2020-10-06T16:33:19Z date_updated: 2023-08-10T08:44:55Z ddc: - '510' department: - _id: '636' doi: 10.3934/jgm.2022014 external_id: arxiv: - '2006.14172' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2022-06-13T09:11:38Z date_updated: 2022-06-13T09:11:38Z description: |- In backward error analysis, an approximate solution to an equa- tion is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting appli- cability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation. file_id: '31859' file_name: 2_BlendedBEASymmPDE.pdf file_size: 1507248 relation: main_file title: Backward error analysis for variational discretisations of PDEs file_date_updated: 2022-06-13T09:11:38Z has_accepted_license: '1' intvolume: ' 14' issue: '3' language: - iso: eng oa: '1' page: 447 - 471 publication: Journal of Geometric Mechanics publication_status: published publisher: AIMS related_material: link: - relation: software url: https://github.com/Christian-Offen/multisymplectic status: public title: Backward error analysis for variational discretisations of partial differential equations type: journal_article user_id: '85279' volume: 14 year: '2022' ... --- _id: '23382' abstract: - lang: eng text: Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes. 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. Symplectic integration of learned Hamiltonian systems. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2022;32(1). doi:10.1063/5.0065913' apa: 'Offen, C., & Ober-Blöbaum, S. (2022). Symplectic integration of learned Hamiltonian systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(1). https://doi.org/10.1063/5.0065913' bibtex: '@article{Offen_Ober-Blöbaum_2022, title={Symplectic integration of learned Hamiltonian systems}, volume={32(1)}, DOI={10.1063/5.0065913}, journal={Chaos: An Interdisciplinary Journal of Nonlinear Science}, publisher={AIP}, author={Offen, Christian and Ober-Blöbaum, Sina}, year={2022} }' chicago: 'Offen, Christian, and Sina Ober-Blöbaum. “Symplectic Integration of Learned Hamiltonian Systems.” Chaos: An Interdisciplinary Journal of Nonlinear Science 32(1) (2022). https://doi.org/10.1063/5.0065913.' ieee: 'C. Offen and S. Ober-Blöbaum, “Symplectic integration of learned Hamiltonian systems,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 32(1), 2022, doi: 10.1063/5.0065913.' mla: 'Offen, Christian, and Sina Ober-Blöbaum. “Symplectic Integration of Learned Hamiltonian Systems.” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 32(1), AIP, 2022, doi:10.1063/5.0065913.' short: 'C. Offen, S. Ober-Blöbaum, Chaos: An Interdisciplinary Journal of Nonlinear Science 32(1) (2022).' date_created: 2021-08-11T08:24:02Z date_updated: 2023-08-10T08:48:14Z ddc: - '510' department: - _id: '636' doi: 10.1063/5.0065913 external_id: arxiv: - '2108.02492' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2021-12-13T14:56:15Z date_updated: 2021-12-13T14:56:15Z file_id: '28734' file_name: SymplecticShadowIntegration_AIP.pdf file_size: 2285059 relation: main_file file_date_updated: 2021-12-13T14:56:15Z has_accepted_license: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://aip.scitation.org/doi/abs/10.1063/5.0065913 oa: '1' publication: 'Chaos: An Interdisciplinary Journal of Nonlinear Science' publication_status: published publisher: AIP quality_controlled: '1' related_material: link: - description: GitHub relation: software url: https://github.com/Christian-Offen/symplectic-shadow-integration status: public title: Symplectic integration of learned Hamiltonian systems type: journal_article user_id: '85279' volume: 32(1) year: '2022' ... --- _id: '22894' abstract: - lang: eng text: "The first order optimality conditions of optimal control problems (OCPs) can\r\nbe regarded as boundary value problems for Hamiltonian systems. Variational or\r\nsymplectic discretisation methods are classically known for their excellent\r\nlong term behaviour. As boundary value problems are posed on intervals of\r\nfixed, moderate length, it is not immediately clear whether methods can profit\r\nfrom structure preservation in this context. When parameters are present,\r\nsolutions can undergo bifurcations, for instance, two solutions can merge and\r\nannihilate one another as parameters are varied. We will show that generic\r\nbifurcations of an OCP are preserved under discretisation when the OCP is\r\neither directly discretised to a discrete OCP (direct method) or translated\r\ninto a Hamiltonian boundary value problem using first order necessary\r\nconditions of optimality which is then solved using a symplectic integrator\r\n(indirect method). Moreover, certain bifurcations break when a non-symplectic\r\nscheme is used. The general phenomenon is illustrated on the example of a cut\r\nlocus of an ellipsoid." 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. Bifurcation preserving discretisations of optimal control problems. 2021;54(19):334-339. doi:https://doi.org/10.1016/j.ifacol.2021.11.099 apa: 'Offen, C., & Ober-Blöbaum, S. (2021). Bifurcation preserving discretisations of optimal control problems: Vol. 54(19) (pp. 334–339). https://doi.org/10.1016/j.ifacol.2021.11.099' bibtex: '@article{Offen_Ober-Blöbaum_2021, series={IFAC-PapersOnLine}, title={Bifurcation preserving discretisations of optimal control problems}, volume={54(19)}, DOI={https://doi.org/10.1016/j.ifacol.2021.11.099}, author={Offen, Christian and Ober-Blöbaum, Sina}, year={2021}, pages={334–339}, collection={IFAC-PapersOnLine} }' chicago: Offen, Christian, and Sina Ober-Blöbaum. “Bifurcation Preserving Discretisations of Optimal Control Problems.” IFAC-PapersOnLine, 2021. https://doi.org/10.1016/j.ifacol.2021.11.099. ieee: 'C. Offen and S. Ober-Blöbaum, “Bifurcation preserving discretisations of optimal control problems,” vol. 54(19). pp. 334–339, 2021, doi: https://doi.org/10.1016/j.ifacol.2021.11.099.' mla: Offen, Christian, and Sina Ober-Blöbaum. Bifurcation Preserving Discretisations of Optimal Control Problems. 2021, pp. 334–39, doi:https://doi.org/10.1016/j.ifacol.2021.11.099. short: C. Offen, S. Ober-Blöbaum, 54(19) (2021) 334–339. conference: end_date: 2021-10-13 location: Berlin, Germany name: 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, LHMNC 2021 start_date: 2021-10-11 date_created: 2021-07-29T09:38:32Z date_updated: 2023-11-29T10:19:41Z ddc: - '510' department: - _id: '636' doi: https://doi.org/10.1016/j.ifacol.2021.11.099 external_id: arxiv: - '2107.13853' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2021-07-29T09:37:49Z date_updated: 2021-07-29T09:37:49Z file_id: '22895' file_name: ifacconf.pdf file_size: 3125220 relation: main_file file_date_updated: 2021-07-29T09:37:49Z has_accepted_license: '1' keyword: - optimal control - catastrophe theory - bifurcations - variational methods - symplectic integrators language: - iso: eng main_file_link: - open_access: '1' url: https://www.sciencedirect.com/science/article/pii/S2405896321021236 oa: '1' page: 334-339 publication_identifier: issn: - 2405-8963 publication_status: published quality_controlled: '1' related_material: link: - description: GitHub/Zenodo relation: software url: https://doi.org/10.5281/zenodo.4562664 series_title: IFAC-PapersOnLine status: public title: Bifurcation preserving discretisations of optimal control problems type: conference user_id: '15694' volume: 54(19) year: '2021' ... --- _id: '21572' author: - first_name: Steffen full_name: Ridderbusch, Steffen last_name: Ridderbusch - 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: Paul full_name: Goulart, Paul last_name: Goulart citation: ama: 'Ridderbusch S, Offen C, Ober-Blöbaum S, Goulart P. Learning ODE Models with Qualitative Structure Using Gaussian Processes . In: 2021 60th IEEE Conference on Decision and Control (CDC). IEEE; 2021:2896. doi:10.1109/CDC45484.2021.9683426' apa: Ridderbusch, S., Offen, C., Ober-Blöbaum, S., & Goulart, P. (2021). Learning ODE Models with Qualitative Structure Using Gaussian Processes . 2021 60th IEEE Conference on Decision and Control (CDC), 2896. https://doi.org/10.1109/CDC45484.2021.9683426 bibtex: '@inproceedings{Ridderbusch_Offen_Ober-Blöbaum_Goulart_2021, title={Learning ODE Models with Qualitative Structure Using Gaussian Processes }, DOI={10.1109/CDC45484.2021.9683426}, booktitle={2021 60th IEEE Conference on Decision and Control (CDC)}, publisher={IEEE}, author={Ridderbusch, Steffen and Offen, Christian and Ober-Blöbaum, Sina and Goulart, Paul}, year={2021}, pages={2896} }' chicago: Ridderbusch, Steffen, Christian Offen, Sina Ober-Blöbaum, and Paul Goulart. “Learning ODE Models with Qualitative Structure Using Gaussian Processes .” In 2021 60th IEEE Conference on Decision and Control (CDC), 2896. IEEE, 2021. https://doi.org/10.1109/CDC45484.2021.9683426. ieee: 'S. Ridderbusch, C. Offen, S. Ober-Blöbaum, and P. Goulart, “Learning ODE Models with Qualitative Structure Using Gaussian Processes ,” in 2021 60th IEEE Conference on Decision and Control (CDC), Austin, TX, USA, 2021, p. 2896, doi: 10.1109/CDC45484.2021.9683426.' mla: Ridderbusch, Steffen, et al. “Learning ODE Models with Qualitative Structure Using Gaussian Processes .” 2021 60th IEEE Conference on Decision and Control (CDC), IEEE, 2021, p. 2896, doi:10.1109/CDC45484.2021.9683426. short: 'S. Ridderbusch, C. Offen, S. Ober-Blöbaum, P. Goulart, in: 2021 60th IEEE Conference on Decision and Control (CDC), IEEE, 2021, p. 2896.' conference: end_date: 2021-12-17 location: Austin, TX, USA name: 60th IEEE Conference on Decision and Control (CDC) start_date: 2021-12-14 date_created: 2021-03-30T10:27:44Z date_updated: 2023-11-29T10:24:55Z department: - _id: '636' doi: 10.1109/CDC45484.2021.9683426 external_id: arxiv: - '2011.05364' language: - iso: eng page: '2896' publication: 2021 60th IEEE Conference on Decision and Control (CDC) publication_identifier: eisbn: - 978-1-6654-3659-5 publication_status: published publisher: IEEE related_material: link: - description: GitHub relation: software url: https://github.com/Crown421/StructureGPs-paper status: public title: 'Learning ODE Models with Qualitative Structure Using Gaussian Processes ' type: conference user_id: '15694' year: '2021' ... --- _id: '19938' abstract: - lang: eng text: 'We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error. ' article_type: original author: - first_name: Robert I full_name: McLachlan, Robert I last_name: McLachlan - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: https://orcid.org/0000-0002-5940-8057 citation: ama: McLachlan RI, Offen C. Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation. Foundations of Computational Mathematics. 2020;20(6):1363-1400. doi:10.1007/s10208-020-09454-z apa: McLachlan, R. I., & Offen, C. (2020). Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation. Foundations of Computational Mathematics, 20(6), 1363–1400. https://doi.org/10.1007/s10208-020-09454-z bibtex: '@article{McLachlan_Offen_2020, title={Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}, volume={20}, DOI={10.1007/s10208-020-09454-z}, number={6}, journal={Foundations of Computational Mathematics}, author={McLachlan, Robert I and Offen, Christian}, year={2020}, pages={1363–1400} }' chicago: 'McLachlan, Robert I, and Christian Offen. “Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation.” Foundations of Computational Mathematics 20, no. 6 (2020): 1363–1400. https://doi.org/10.1007/s10208-020-09454-z.' ieee: R. I. McLachlan and C. Offen, “Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation,” Foundations of Computational Mathematics, vol. 20, no. 6, pp. 1363–1400, 2020. mla: McLachlan, Robert I., and Christian Offen. “Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation.” Foundations of Computational Mathematics, vol. 20, no. 6, 2020, pp. 1363–400, doi:10.1007/s10208-020-09454-z. short: R.I. McLachlan, C. Offen, Foundations of Computational Mathematics 20 (2020) 1363–1400. date_created: 2020-10-06T16:31:46Z date_updated: 2022-01-06T06:54:14Z department: - _id: '636' doi: 10.1007/s10208-020-09454-z extern: '1' intvolume: ' 20' issue: '6' language: - iso: eng main_file_link: - url: https://rdcu.be/b79aB page: 1363-1400 publication: Foundations of Computational Mathematics publication_status: published status: public title: Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation type: journal_article user_id: '85279' volume: 20 year: '2020' ... --- _id: '19939' article_type: original author: - first_name: Lisa Maria full_name: Kreusser, Lisa Maria last_name: Kreusser - first_name: Robert I full_name: McLachlan, Robert I last_name: McLachlan - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: https://orcid.org/0000-0002-5940-8057 citation: ama: Kreusser LM, McLachlan RI, Offen C. Detection of high codimensional bifurcations in variational PDEs. Nonlinearity. 2020;33(5):2335-2363. doi:10.1088/1361-6544/ab7293 apa: Kreusser, L. M., McLachlan, R. I., & Offen, C. (2020). Detection of high codimensional bifurcations in variational PDEs. Nonlinearity, 33(5), 2335–2363. https://doi.org/10.1088/1361-6544/ab7293 bibtex: '@article{Kreusser_McLachlan_Offen_2020, title={Detection of high codimensional bifurcations in variational PDEs}, volume={33}, DOI={10.1088/1361-6544/ab7293}, number={5}, journal={Nonlinearity}, author={Kreusser, Lisa Maria and McLachlan, Robert I and Offen, Christian}, year={2020}, pages={2335–2363} }' chicago: 'Kreusser, Lisa Maria, Robert I McLachlan, and Christian Offen. “Detection of High Codimensional Bifurcations in Variational PDEs.” Nonlinearity 33, no. 5 (2020): 2335–63. https://doi.org/10.1088/1361-6544/ab7293.' ieee: L. M. Kreusser, R. I. McLachlan, and C. Offen, “Detection of high codimensional bifurcations in variational PDEs,” Nonlinearity, vol. 33, no. 5, pp. 2335–2363, 2020. mla: Kreusser, Lisa Maria, et al. “Detection of High Codimensional Bifurcations in Variational PDEs.” Nonlinearity, vol. 33, no. 5, 2020, pp. 2335–63, doi:10.1088/1361-6544/ab7293. short: L.M. Kreusser, R.I. McLachlan, C. Offen, Nonlinearity 33 (2020) 2335–2363. date_created: 2020-10-06T16:32:04Z date_updated: 2022-01-06T06:54:14Z department: - _id: '636' doi: 10.1088/1361-6544/ab7293 extern: '1' intvolume: ' 33' issue: '5' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.1088/1361-6544/ab7293 oa: '1' page: 2335-2363 publication: Nonlinearity publication_identifier: issn: - 0951-7715 - 1361-6544 publication_status: published status: public title: Detection of high codimensional bifurcations in variational PDEs type: journal_article user_id: '85279' volume: 33 year: '2020' ... --- _id: '19947' abstract: - lang: eng text: "Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise\r\nin most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate\r\nsolutions. In order to draw valid conclusions from numerical computations, it is crucial\r\nto understand which qualitative aspects numerical solutions have in common with the\r\nexact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity\r\nunder discretisation on long-term behaviour of motions is classically well known, in this\r\nthesis\r\n(a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian\r\nboundary value problems is explained. In parameter dependent systems at a bifurcation\r\npoint the solution set to a boundary value problem changes qualitatively. Bifurcation\r\nproblems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to\r\npersistent bifurcations of Hamiltonian boundary value problems. Further results for\r\nsymmetric settings are derived.\r\n(b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem.\r\n(c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs\r\nwith variational structure. Recognition equations for A-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations.\r\n(d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers’ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed.\r\nIt is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation.\r\n(e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating)\r\ntravelling waves in the nonlinear wave equation is discussed." alternative_title: - A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand. author: - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: https://orcid.org/0000-0002-5940-8057 citation: ama: 'Offen C. Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration. Palmerston North, New Zealand: Massey University; 2020.' apa: 'Offen, C. (2020). Analysis of Hamiltonian boundary value problems and symplectic integration. Palmerston North, New Zealand: Massey University.' bibtex: '@book{Offen_2020, place={Palmerston North, New Zealand}, title={Analysis of Hamiltonian boundary value problems and symplectic integration}, publisher={Massey University}, author={Offen, Christian}, year={2020} }' chicago: 'Offen, Christian. Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration. Palmerston North, New Zealand: Massey University, 2020.' ieee: 'C. Offen, Analysis of Hamiltonian boundary value problems and symplectic integration. Palmerston North, New Zealand: Massey University, 2020.' mla: Offen, Christian. Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration. Massey University, 2020. short: C. Offen, Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration, Massey University, Palmerston North, New Zealand, 2020. date_created: 2020-10-06T18:56:44Z date_updated: 2022-01-06T06:54:16Z ddc: - '510' extern: '1' file: - access_level: open_access content_type: application/pdf creator: coffen date_created: 2020-10-06T18:54:53Z date_updated: 2020-10-07T14:01:58Z description: |- A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand. file_id: '19948' file_name: ths_all_signatures.pdf file_size: 19465740 relation: main_file title: Thesis Christian Offen file_date_updated: 2020-10-07T14:01:58Z has_accepted_license: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://hdl.handle.net/10179/16155 oa: '1' place: Palmerston North, New Zealand publication_status: published publisher: Massey University status: public supervisor: - first_name: Robert I full_name: McLachlan, Robert I last_name: McLachlan title: Analysis of Hamiltonian boundary value problems and symplectic integration type: dissertation user_id: '85279' year: '2020' ... --- _id: '19945' abstract: - lang: eng text: Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration. article_type: original author: - first_name: Robert I full_name: McLachlan, Robert I last_name: McLachlan - first_name: Christian full_name: Offen, Christian id: '85279' last_name: Offen orcid: https://orcid.org/0000-0002-5940-8057 - first_name: Benjamin K full_name: Tapley, Benjamin K last_name: Tapley citation: ama: McLachlan RI, Offen C, Tapley BK. Symplectic integration of PDEs using Clebsch variables. Journal of Computational Dynamics. 2019;6(1):111-130. doi:10.3934/jcd.2019005 apa: McLachlan, R. I., Offen, C., & Tapley, B. K. (2019). Symplectic integration of PDEs using Clebsch variables. Journal of Computational Dynamics, 6(1), 111–130. https://doi.org/10.3934/jcd.2019005 bibtex: '@article{McLachlan_Offen_Tapley_2019, title={Symplectic integration of PDEs using Clebsch variables}, volume={6}, DOI={10.3934/jcd.2019005}, number={1}, journal={Journal of Computational Dynamics}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={McLachlan, Robert I and Offen, Christian and Tapley, Benjamin K}, year={2019}, pages={111–130} }' chicago: 'McLachlan, Robert I, Christian Offen, and Benjamin K Tapley. “Symplectic Integration of PDEs Using Clebsch Variables.” Journal of Computational Dynamics 6, no. 1 (2019): 111–30. https://doi.org/10.3934/jcd.2019005.' ieee: R. I. McLachlan, C. Offen, and B. K. Tapley, “Symplectic integration of PDEs using Clebsch variables,” Journal of Computational Dynamics, vol. 6, no. 1, pp. 111–130, 2019. mla: McLachlan, Robert I., et al. “Symplectic Integration of PDEs Using Clebsch Variables.” Journal of Computational Dynamics, vol. 6, no. 1, American Institute of Mathematical Sciences (AIMS), 2019, pp. 111–30, doi:10.3934/jcd.2019005. short: R.I. McLachlan, C. Offen, B.K. Tapley, Journal of Computational Dynamics 6 (2019) 111–130. date_created: 2020-10-06T16:44:07Z date_updated: 2022-01-06T06:54:15Z department: - _id: '636' doi: 10.3934/jcd.2019005 extern: '1' intvolume: ' 6' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: http://www.aimsciences.org/article/doi/10.3934/jcd.2019005 oa: '1' page: 111-130 publication: Journal of Computational Dynamics publication_identifier: issn: - 2158-2505 publisher: American Institute of Mathematical Sciences (AIMS) status: public title: Symplectic integration of PDEs using Clebsch variables type: journal_article user_id: '85279' volume: 6 year: '2019' ...