[{"user_id":"85279","ddc":["510"],"article_type":"original","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. "}],"status":"public","has_accepted_license":"1","date_created":"2023-08-10T08:24:48Z","volume":34,"file":[{"file_size":13222105,"creator":"coffen","file_id":"50376","title":"Accepted Manuscript Chaos","date_updated":"2024-01-09T10:48:38Z","content_type":"application/pdf","relation":"main_file","date_created":"2024-01-09T10:48:38Z","file_name":"Accepted manuscript with AIP banner CHA23-AR-01370.pdf","access_level":"open_access"},{"access_level":"open_access","date_created":"2024-01-09T11:19:49Z","file_name":"LDensityPDE_AIP.pdf","content_type":"application/pdf","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\na neural network model of a discrete Lagrangian density such that the discrete Euler–Lagrange equations are consistent\nwith the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian\ndensities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for\nthe training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers\nguarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical\nsimulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory\nsuch as travelling waves. This is possible even when travelling waves are not present in the training data. This is\ncompared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces\ncontaining structurally simple solutions when these are not present in the training data. Ideas are demonstrated on\nexamples based on the wave equation and the Schrödinger equation.","relation":"main_file","file_size":12960884,"title":"Learning of discrete models of variational PDEs from data","file_id":"50390","creator":"coffen"}],"publisher":"AIP Publishing","quality_controlled":"1","author":[{"first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian","last_name":"Offen","id":"85279"},{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"}],"publication":"Chaos","file_date_updated":"2024-01-09T11:19:49Z","issue":"1","article_number":"013104","_id":"46469","intvolume":" 34","year":"2024","citation":{"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.","short":"C. Offen, S. Ober-Blöbaum, Chaos 34 (2024).","bibtex":"@article{Offen_Ober-Blöbaum_2024, title={Learning of discrete models of variational PDEs from data}, volume={34}, DOI={10.1063/5.0172287}, number={1013104}, journal={Chaos}, publisher={AIP Publishing}, author={Offen, Christian and Ober-Blöbaum, Sina}, year={2024} }","mla":"Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational PDEs from Data.” Chaos, vol. 34, no. 1, 013104, AIP Publishing, 2024, doi:10.1063/5.0172287.","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.","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"},"type":"journal_article","related_material":{"link":[{"description":"GitHub","relation":"software","url":"https://github.com/Christian-Offen/DLNN_pde"}]},"title":"Learning of discrete models of variational PDEs from data","external_id":{"arxiv":["2308.05082 "]},"publication_identifier":{"issn":["1054-1500"]},"publication_status":"published","department":[{"_id":"636"}],"oa":"1","doi":"10.1063/5.0172287","date_updated":"2024-01-09T11:29:06Z","language":[{"iso":"eng"}]},{"issue":"0","_id":"53101","citation":{"mla":"Leyendecker, Sigrid, et al. “A New Lagrangian Approach to Control Affine Systems with a Quadratic Lagrange Term.” Journal of Computational Dynamics, vol. 0, no. 0, American Institute of Mathematical Sciences (AIMS), 2024, pp. 0–0, doi:10.3934/jcd.2024017.","bibtex":"@article{Leyendecker_Maslovskaya_Ober-Blöbaum_Almagro_Szemenyei_2024, title={A new Lagrangian approach to control affine systems with a quadratic Lagrange term}, volume={0}, DOI={10.3934/jcd.2024017}, number={0}, journal={Journal of Computational Dynamics}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Almagro, Rodrigo T. Sato Martín de and Szemenyei, Flóra Orsolya}, year={2024}, pages={0–0} }","apa":"Leyendecker, S., Maslovskaya, S., Ober-Blöbaum, S., Almagro, R. T. S. M. de, & Szemenyei, F. O. (2024). A new Lagrangian approach to control affine systems with a quadratic Lagrange term. Journal of Computational Dynamics, 0(0), 0–0. https://doi.org/10.3934/jcd.2024017","ama":"Leyendecker S, Maslovskaya S, Ober-Blöbaum S, Almagro RTSM de, Szemenyei FO. A new Lagrangian approach to control affine systems with a quadratic Lagrange term. Journal of Computational Dynamics. 2024;0(0):0-0. doi:10.3934/jcd.2024017","chicago":"Leyendecker, Sigrid, Sofya Maslovskaya, Sina Ober-Blöbaum, Rodrigo T. Sato Martín de Almagro, and Flóra Orsolya Szemenyei. “A New Lagrangian Approach to Control Affine Systems with a Quadratic Lagrange Term.” Journal of Computational Dynamics 0, no. 0 (2024): 0–0. https://doi.org/10.3934/jcd.2024017.","ieee":"S. Leyendecker, S. Maslovskaya, S. Ober-Blöbaum, R. T. S. M. de Almagro, and F. O. Szemenyei, “A new Lagrangian approach to control affine systems with a quadratic Lagrange term,” Journal of Computational Dynamics, vol. 0, no. 0, pp. 0–0, 2024, doi: 10.3934/jcd.2024017.","short":"S. Leyendecker, S. Maslovskaya, S. Ober-Blöbaum, R.T.S.M. de Almagro, F.O. Szemenyei, Journal of Computational Dynamics 0 (2024) 0–0."},"year":"2024","type":"journal_article","page":"0-0","main_file_link":[{"open_access":"1","url":"https://www.aimsciences.org/article/doi/10.3934/jcd.2024017"}],"ddc":["510"],"user_id":"87909","article_type":"original","abstract":[{"text":"In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.","lang":"eng"}],"volume":"0","status":"public","has_accepted_license":"1","date_created":"2024-03-28T15:58:02Z","publisher":"American Institute of Mathematical Sciences (AIMS)","author":[{"first_name":"Sigrid","full_name":"Leyendecker, Sigrid","last_name":"Leyendecker"},{"last_name":"Maslovskaya","id":"87909","first_name":"Sofya","full_name":"Maslovskaya, Sofya"},{"id":"16494","last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina","first_name":"Sina"},{"full_name":"Almagro, Rodrigo T. Sato Martín de","first_name":"Rodrigo T. Sato Martín de","last_name":"Almagro"},{"first_name":"Flóra Orsolya","full_name":"Szemenyei, Flóra Orsolya","last_name":"Szemenyei"}],"publication":"Journal of Computational Dynamics","keyword":["Optimal control problem","Lagrangian system","Hamiltonian system","Variations","Pontryagin's maximum principle."],"doi":"10.3934/jcd.2024017","oa":"1","date_updated":"2024-03-28T16:07:34Z","language":[{"iso":"eng"}],"title":"A new Lagrangian approach to control affine systems with a quadratic Lagrange term","publication_identifier":{"issn":["2158-2491","2158-2505"]},"publication_status":"published","department":[{"_id":"636"}]},{"series_title":"Lecture Notes in Computer Science (LNCS)","language":[{"iso":"eng"}],"date_updated":"2023-08-10T08:34:04Z","oa":"1","doi":"10.1007/978-3-031-38271-0_57","department":[{"_id":"636"}],"editor":[{"last_name":"Nielsen","full_name":"Nielsen, F","first_name":"F"},{"first_name":"F","full_name":"Barbaresco, F","last_name":"Barbaresco"}],"publication_status":"published","publication_identifier":{"eisbn":["978-3-031-38271-0"]},"external_id":{"arxiv":["2302.08232 "]},"related_material":{"link":[{"description":"GitHub","relation":"software","url":"https://github.com/Christian-Offen/LagrangianDensityML"}]},"title":"Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves","type":"conference","year":"2023","citation":{"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.","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","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","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)} }","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.","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."},"page":"569-579","intvolume":" 14071","_id":"42163","conference":{"location":"Saint-Malo, Palais du Grand Large, France","start_date":"2023-08-30","name":" GSI'23 6th International Conference on Geometric Science of Information","end_date":"2023-09-01"},"file":[{"access_level":"open_access","file_name":"LDensityLearning.pdf","date_created":"2023-08-02T12:04:17Z","relation":"main_file","description":"The article shows how to learn models of dynamical systems\nfrom data which are governed by an unknown variational PDE. Rather\nthan employing reduction techniques, we learn a discrete field theory\ngoverned by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is\nnecessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of\nthe discrete Euler–Lagrange equations. Secondly, we develop a method to\nfind solutions to machine learned discrete field theories which constitute\ntravelling waves of the underlying continuous PDE.","date_updated":"2023-08-02T12:04:17Z","content_type":"application/pdf","file_id":"46273","creator":"coffen","title":"Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves","file_size":1938962}],"author":[{"id":"85279","last_name":"Offen","full_name":"Offen, Christian","orcid":"0000-0002-5940-8057","first_name":"Christian"},{"full_name":"Ober-Blöbaum, Sina","first_name":"Sina","id":"16494","last_name":"Ober-Blöbaum"}],"quality_controlled":"1","publisher":"Springer, Cham.","file_date_updated":"2023-08-02T12:04:17Z","keyword":["System identification","discrete Lagrangians","travelling waves"],"publication":"Geometric Science of Information","has_accepted_license":"1","status":"public","date_created":"2023-02-16T11:32:48Z","volume":14071,"abstract":[{"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.","lang":"eng"}],"user_id":"85279","ddc":["510"]},{"department":[{"_id":"636"}],"publication_identifier":{"issn":["0377-0427"]},"publication_status":"epub_ahead","external_id":{"arxiv":["2112.12619"]},"title":"Variational Learning of Euler–Lagrange Dynamics from Data","related_material":{"link":[{"relation":"software","url":"https://github.com/Christian-Offen/LagrangianShadowIntegration"}]},"language":[{"iso":"eng"}],"date_updated":"2023-08-10T08:42:39Z","doi":"10.1016/j.cam.2022.114780","oa":"1","quality_controlled":"1","publisher":"Elsevier","author":[{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"},{"first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian","last_name":"Offen","id":"85279"}],"file_date_updated":"2022-06-28T15:25:50Z","publication":"Journal of Computational and Applied Mathematics","keyword":["Lagrangian learning","variational backward error analysis","modified Lagrangian","variational integrators","physics informed learning"],"file":[{"title":"Variational Learning of Euler–Lagrange Dynamics from Data","file_size":3640770,"file_name":"ShadowLagrangian_revision1_journal_style_arxiv.pdf","date_created":"2022-06-28T15:25:50Z","access_level":"open_access","file_id":"32274","creator":"coffen","description":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions\nconnecting two points in a phase space, the system will exhibit those motions which extremise an action functional.\nMany qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equa-\ntions, are related to the existence of an action functional. Incorporating variational structure into learning algorithms\nfor dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features\nwith the exact physical system. In this paper we show how to incorporate variational principles into trajectory predic-\ntions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position\ndata of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no\nprior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward\nerror analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the\nlearned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of\nvariational backward error analysis. (3) Finally, we introduce a method to perform system identification from position\nobservations only, based on variational backward error analysis.","relation":"main_file","date_updated":"2022-06-28T15:25:50Z","content_type":"application/pdf"}],"volume":421,"has_accepted_license":"1","status":"public","date_created":"2022-01-11T13:24:00Z","article_type":"original","abstract":[{"text":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\r\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.","lang":"eng"}],"ddc":["510"],"user_id":"85279","citation":{"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.","short":"S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics 421 (2023) 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} }","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.","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.","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"},"type":"journal_article","year":"2023","page":"114780","_id":"29240","intvolume":" 421"},{"user_id":"85279","ddc":["510"],"article_type":"original","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."}],"has_accepted_license":"1","status":"public","date_created":"2023-01-20T09:10:06Z","volume":33,"file":[{"file_id":"44205","creator":"coffen","relation":"main_file","description":"Incorporating physical system knowledge into data-driven\nsystem identification has been shown to be beneficial. The\napproach presented in this article combines learning of an\nenergy-conserving model from data with detecting a Lie\ngroup representation of the unknown system symmetry.\nThe proposed approach can improve the learned model\nand reveal underlying symmetry simultaneously.","content_type":"application/pdf","date_updated":"2023-04-26T16:20:56Z","date_created":"2023-04-26T16:20:56Z","file_name":"JournalPaper_main.pdf","access_level":"open_access","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","file_size":5200111}],"author":[{"last_name":"Dierkes","first_name":"Eva","full_name":"Dierkes, Eva"},{"last_name":"Offen","id":"85279","first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian"},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"},{"last_name":"Flaßkamp","full_name":"Flaßkamp, Kathrin","first_name":"Kathrin"}],"publisher":"AIP Publishing","publication":"Chaos","file_date_updated":"2023-04-26T16:20:56Z","issue":"6","article_number":"063115","intvolume":" 33","_id":"37654","year":"2023","citation":{"short":"E. Dierkes, C. Offen, S. Ober-Blöbaum, K. Flaßkamp, Chaos 33 (2023).","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.","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.","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} }","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."},"type":"journal_article","related_material":{"link":[{"description":"GitHub","relation":"software","url":"https://github.com/eva-dierkes/HNN_withSymmetries"}]},"title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","external_id":{"arxiv":["2301.07928"]},"publication_identifier":{"issn":["1054-1500"]},"publication_status":"published","department":[{"_id":"636"}],"oa":"1","doi":"10.1063/5.0142969","date_updated":"2023-08-10T08:37:01Z","language":[{"iso":"eng"}]},{"department":[{"_id":"101"},{"_id":"636"},{"_id":"355"},{"_id":"655"}],"publication_status":"published","external_id":{"arxiv":["arXiv:2104.03562"]},"related_material":{"link":[{"url":"https://github.com/lueckem/quadrature-ML","description":"GitHub","relation":"software"}]},"title":"Efficient time stepping for numerical integration using reinforcement learning","language":[{"iso":"eng"}],"date_updated":"2023-08-25T09:24:50Z","doi":"10.1137/21M1412682","author":[{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"},{"last_name":"Hüllermeier","id":"48129","first_name":"Eyke","full_name":"Hüllermeier, Eyke"},{"first_name":"Marvin","full_name":"Lücke, Marvin","last_name":"Lücke"},{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"},{"first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian","last_name":"Offen","id":"85279"},{"first_name":"Sebastian","orcid":"0000-0002-3389-793X","full_name":"Peitz, Sebastian","last_name":"Peitz","id":"47427"},{"id":"13472","last_name":"Pfannschmidt","full_name":"Pfannschmidt, Karlson","orcid":"0000-0001-9407-7903","first_name":"Karlson"}],"publication":"SIAM Journal on Scientific Computing","status":"public","has_accepted_license":"1","date_created":"2021-04-09T07:59:19Z","volume":45,"abstract":[{"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.","lang":"eng"}],"user_id":"47427","ddc":["510"],"main_file_link":[{"url":"https://epubs.siam.org/doi/reader/10.1137/21M1412682"}],"type":"journal_article","year":"2023","citation":{"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.","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","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","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} }","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.","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."},"page":"A579-A595","_id":"21600","intvolume":" 45","issue":"2"},{"user_id":"15694","title":"Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model ","status":"public","date_created":"2023-09-21T07:20:43Z","alternative_title":["GAMM Rundbriefe"],"publisher":"Invited Mathematical Magazine Article","author":[{"last_name":"Lishkova","first_name":"Yana","full_name":"Lishkova, Yana"},{"full_name":"Ober-Blöbaum, Sina","first_name":"Sina","id":"16494","last_name":"Ober-Blöbaum"},{"last_name":"Leyendecker","full_name":"Leyendecker, Sigrid","first_name":"Sigrid"}],"_id":"47147","date_updated":"2023-09-21T07:27:15Z","language":[{"iso":"eng"}],"year":"2023","citation":{"mla":"Lishkova, Yana, et al. Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model . Invited Mathematical Magazine Article, 2023.","bibtex":"@book{Lishkova_Ober-Blöbaum_Leyendecker_2023, title={Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model }, publisher={Invited Mathematical Magazine Article}, author={Lishkova, Yana and Ober-Blöbaum, Sina and Leyendecker, Sigrid}, year={2023} }","apa":"Lishkova, Y., Ober-Blöbaum, S., & Leyendecker, S. (2023). Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model . Invited Mathematical Magazine Article.","ama":"Lishkova Y, Ober-Blöbaum S, Leyendecker S. Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model . Invited Mathematical Magazine Article; 2023.","chicago":"Lishkova, Yana, Sina Ober-Blöbaum, and Sigrid Leyendecker. Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model . Invited Mathematical Magazine Article, 2023.","ieee":"Y. Lishkova, S. Ober-Blöbaum, and S. Leyendecker, Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model . Invited Mathematical Magazine Article, 2023.","short":"Y. Lishkova, S. Ober-Blöbaum, S. Leyendecker, Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model , Invited Mathematical Magazine Article, 2023."},"type":"report"},{"user_id":"15694","title":"Variational approach for modelling and optimal control of electrodynamic tether motion","status":"public","date_created":"2023-09-21T07:25:40Z","publisher":"Accepted for the 74th International Astronautical Concress (IAC)","author":[{"full_name":"Lishkova, Y.","first_name":"Y.","last_name":"Lishkova"},{"last_name":"Bando","full_name":"Bando, M.","first_name":"M."},{"id":"16494","last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina","first_name":"Sina"}],"_id":"47148","date_updated":"2023-09-21T07:27:21Z","language":[{"iso":"eng"}],"citation":{"short":"Y. Lishkova, M. Bando, S. Ober-Blöbaum, in: Accepted for the 74th International Astronautical Concress (IAC), 2023.","ieee":"Y. Lishkova, M. Bando, and S. Ober-Blöbaum, “Variational approach for modelling and optimal control of electrodynamic tether motion,” 2023.","chicago":"Lishkova, Y., M. Bando, and Sina Ober-Blöbaum. “Variational Approach for Modelling and Optimal Control of Electrodynamic Tether Motion.” Accepted for the 74th International Astronautical Concress (IAC), 2023.","ama":"Lishkova Y, Bando M, Ober-Blöbaum S. Variational approach for modelling and optimal control of electrodynamic tether motion. In: Accepted for the 74th International Astronautical Concress (IAC); 2023.","apa":"Lishkova, Y., Bando, M., & Ober-Blöbaum, S. (2023). Variational approach for modelling and optimal control of electrodynamic tether motion.","mla":"Lishkova, Y., et al. Variational Approach for Modelling and Optimal Control of Electrodynamic Tether Motion. Accepted for the 74th International Astronautical Concress (IAC), 2023.","bibtex":"@inproceedings{Lishkova_Bando_Ober-Blöbaum_2023, title={Variational approach for modelling and optimal control of electrodynamic tether motion}, publisher={Accepted for the 74th International Astronautical Concress (IAC)}, author={Lishkova, Y. and Bando, M. and Ober-Blöbaum, Sina}, year={2023} }"},"year":"2023","type":"conference"},{"publication_status":"published","department":[{"_id":"636"}],"related_material":{"link":[{"url":"https://github.com/yanalish/SymDLNN","relation":"software","description":"GitHub"}]},"title":"Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery","external_id":{"arxiv":["2211.10830"]},"language":[{"iso":"eng"}],"oa":"1","doi":"10.1016/j.ifacol.2023.10.1457","date_updated":"2023-12-29T14:26:00Z","has_accepted_license":"1","status":"public","date_created":"2022-11-23T08:17:10Z","volume":56,"file":[{"date_updated":"2023-04-17T08:05:55Z","content_type":"application/pdf","description":"By one of the most fundamental principles in physics, a dynamical system will\nexhibit those motions which extremise an action functional. This leads to the formation of\nthe Euler-Lagrange equations, which serve as a model of how the system will behave in time.\nIf the dynamics exhibit additional symmetries, then the motion fulfils additional conservation\nlaws, such as conservation of energy (time invariance), momentum (translation invariance), or\nangular momentum (rotational invariance). To learn a system representation, one could learn\nthe discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function\nLd which defines them. Based on ideas from Lie group theory, we introduce a framework to learn\na discrete Lagrangian along with its symmetry group from discrete observations of motions and,\ntherefore, identify conserved quantities. The learning process does not restrict the form of the\nLagrangian, does not require velocity or momentum observations or predictions and incorporates\na cost term which safeguards against unwanted solutions and against potential numerical issues\nin forward simulations. The learnt discrete quantities are related to their continuous analogues\nusing variational backward error analysis and numerical results demonstrate the improvement\nsuch models can have both qualitatively and quantitatively even in the presence of noise.","relation":"main_file","file_id":"44037","creator":"coffen","file_size":576115,"title":"Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery","access_level":"open_access","date_created":"2023-04-17T08:05:55Z","file_name":"LNN_project.pdf"}],"quality_controlled":"1","author":[{"first_name":"Yana","full_name":"Lishkova, Yana","last_name":"Lishkova"},{"full_name":"Scherer, Paul","first_name":"Paul","last_name":"Scherer"},{"full_name":"Ridderbusch, Steffen","first_name":"Steffen","last_name":"Ridderbusch"},{"first_name":"Mateja","full_name":"Jamnik, Mateja","last_name":"Jamnik"},{"last_name":"Liò","full_name":"Liò, Pietro","first_name":"Pietro"},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"},{"first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian","last_name":"Offen","id":"85279"}],"publisher":"Elsevier","publication":"IFAC-PapersOnLine","file_date_updated":"2023-04-17T08:05:55Z","user_id":"85279","ddc":["510"],"abstract":[{"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.","lang":"eng"}],"year":"2023","type":"conference","citation":{"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.","short":"Y. Lishkova, P. Scherer, S. Ridderbusch, M. Jamnik, P. Liò, S. Ober-Blöbaum, C. Offen, in: IFAC-PapersOnLine, Elsevier, 2023, pp. 3203–3210.","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.","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} }","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","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."},"page":"3203-3210","main_file_link":[{"url":"https://www.sciencedirect.com/science/article/pii/S2405896323018657"}],"issue":"2","_id":"34135","intvolume":" 56","conference":{"location":" Yokohama, Japan","start_date":"2023-07-09","name":"The 22nd World Congress of the International Federation of Automatic Control","end_date":"2023-07-14"}},{"citation":{"mla":"Cresson, Jacky, et al. “Continuous and Discrete Noether’s Fractional Conserved Quantities for Restricted Calculus of Variations.” AIMS, vol. 14(1), 2022, pp. 57–89.","bibtex":"@article{Cresson_Jiménez_Ober-Blöbaum_2022, title={Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations}, volume={14(1)}, journal={AIMS}, author={Cresson, Jacky and Jiménez, Fernando and Ober-Blöbaum, Sina}, year={2022}, pages={57–89} }","ieee":"J. Cresson, F. Jiménez, and S. Ober-Blöbaum, “Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations,” AIMS, vol. 14(1), pp. 57–89, 2022.","chicago":"Cresson, Jacky, Fernando Jiménez, and Sina Ober-Blöbaum. “Continuous and Discrete Noether’s Fractional Conserved Quantities for Restricted Calculus of Variations.” AIMS 14(1) (2022): 57–89.","ama":"Cresson J, Jiménez F, Ober-Blöbaum S. Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations. AIMS. 2022;14(1):57-89.","short":"J. Cresson, F. Jiménez, S. Ober-Blöbaum, AIMS 14(1) (2022) 57–89.","apa":"Cresson, J., Jiménez, F., & Ober-Blöbaum, S. (2022). Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations. AIMS, 14(1), 57–89."},"year":"2022","type":"journal_article","page":"57-89","language":[{"iso":"eng"}],"_id":"30490","date_updated":"2022-03-24T12:26:32Z","volume":"14(1)","status":"public","date_created":"2022-03-24T12:26:10Z","author":[{"full_name":"Cresson, Jacky","first_name":"Jacky","last_name":"Cresson"},{"last_name":"Jiménez","full_name":"Jiménez, Fernando","first_name":"Fernando"},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"}],"department":[{"_id":"636"}],"publication":"AIMS","title":"Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations","user_id":"15694"},{"publication":"Mathematics of Control, Signals, and Systems","department":[{"_id":"636"}],"author":[{"first_name":"Timm","full_name":"Faulwasser, Timm","last_name":"Faulwasser"},{"last_name":"Flaßkamp","first_name":"Kathrin","full_name":"Flaßkamp, Kathrin"},{"full_name":"Ober-Blöbaum, Sina","first_name":"Sina","id":"16494","last_name":"Ober-Blöbaum"},{"full_name":"Schaller, Manuel","first_name":"Manuel","last_name":"Schaller"},{"full_name":"Worthmann, Karl","first_name":"Karl","last_name":"Worthmann"}],"publisher":"Springer","volume":34,"date_created":"2023-05-08T09:04:06Z","status":"public","title":"Manifold turnpikes, trims, and symmetries","user_id":"15694","page":"759-788","citation":{"chicago":"Faulwasser, Timm, Kathrin Flaßkamp, Sina Ober-Blöbaum, Manuel Schaller, and Karl Worthmann. “Manifold Turnpikes, Trims, and Symmetries.” Mathematics of Control, Signals, and Systems 34 (2022): 759–88.","apa":"Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S., Schaller, M., & Worthmann, K. (2022). Manifold turnpikes, trims, and symmetries. Mathematics of Control, Signals, and Systems, 34, 759–788.","ama":"Faulwasser T, Flaßkamp K, Ober-Blöbaum S, Schaller M, Worthmann K. Manifold turnpikes, trims, and symmetries. Mathematics of Control, Signals, and Systems. 2022;34:759-788.","bibtex":"@article{Faulwasser_Flaßkamp_Ober-Blöbaum_Schaller_Worthmann_2022, title={Manifold turnpikes, trims, and symmetries}, volume={34}, journal={Mathematics of Control, Signals, and Systems}, publisher={Springer}, author={Faulwasser, Timm and Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Schaller, Manuel and Worthmann, Karl}, year={2022}, pages={759–788} }","mla":"Faulwasser, Timm, et al. “Manifold Turnpikes, Trims, and Symmetries.” Mathematics of Control, Signals, and Systems, vol. 34, Springer, 2022, pp. 759–88.","short":"T. Faulwasser, K. Flaßkamp, S. Ober-Blöbaum, M. Schaller, K. Worthmann, Mathematics of Control, Signals, and Systems 34 (2022) 759–788.","ieee":"T. Faulwasser, K. Flaßkamp, S. Ober-Blöbaum, M. Schaller, and K. Worthmann, “Manifold turnpikes, trims, and symmetries,” Mathematics of Control, Signals, and Systems, vol. 34, pp. 759–788, 2022."},"type":"journal_article","year":"2022","language":[{"iso":"eng"}],"_id":"44624","date_updated":"2023-05-08T09:04:26Z","intvolume":" 34"},{"external_id":{"arxiv":["2108.02492"]},"related_material":{"link":[{"url":"https://github.com/Christian-Offen/symplectic-shadow-integration","description":"GitHub","relation":"software"}]},"title":"Symplectic integration of learned Hamiltonian systems","department":[{"_id":"636"}],"publication_status":"published","date_updated":"2023-08-10T08:48:14Z","oa":"1","doi":"10.1063/5.0065913","language":[{"iso":"eng"}],"article_type":"original","abstract":[{"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.","lang":"eng"}],"user_id":"85279","ddc":["510"],"file":[{"access_level":"open_access","date_created":"2021-12-13T14:56:15Z","file_name":"SymplecticShadowIntegration_AIP.pdf","relation":"main_file","date_updated":"2021-12-13T14:56:15Z","content_type":"application/pdf","creator":"coffen","file_id":"28734","file_size":2285059}],"quality_controlled":"1","publisher":"AIP","author":[{"id":"85279","last_name":"Offen","full_name":"Offen, Christian","orcid":"0000-0002-5940-8057","first_name":"Christian"},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"}],"file_date_updated":"2021-12-13T14:56:15Z","publication":"Chaos: An Interdisciplinary Journal of Nonlinear Science","status":"public","has_accepted_license":"1","date_created":"2021-08-11T08:24:02Z","volume":"32(1)","_id":"23382","main_file_link":[{"url":"https://aip.scitation.org/doi/abs/10.1063/5.0065913","open_access":"1"}],"year":"2022","citation":{"short":"C. Offen, S. Ober-Blöbaum, Chaos: An Interdisciplinary Journal of Nonlinear Science 32(1) (2022).","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.","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.","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} }","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."},"type":"journal_article"},{"date_updated":"2023-11-29T10:00:18Z","_id":"30733","conference":{"name":"2022 European Control Conference (ECC)","start_date":"2022-07-12","location":"London","end_date":"2022-07-15"},"language":[{"iso":"eng"}],"citation":{"ieee":"N. Vertovec, S. Ober-Blöbaum, and K. Margellos, “Verification of safety critical control policies using kernel methods,” London, 2022, pp. 1870–1875.","short":"N. Vertovec, S. Ober-Blöbaum, K. Margellos, in: 2022, pp. 1870–1875.","bibtex":"@inproceedings{Vertovec_Ober-Blöbaum_Margellos_2022, title={Verification of safety critical control policies using kernel methods}, author={Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}, year={2022}, pages={1870–1875} }","mla":"Vertovec, Nikolaus, et al. Verification of Safety Critical Control Policies Using Kernel Methods. 2022, pp. 1870–75.","chicago":"Vertovec, Nikolaus, Sina Ober-Blöbaum, and Kostas Margellos. “Verification of Safety Critical Control Policies Using Kernel Methods,” 1870–75, 2022.","apa":"Vertovec, N., Ober-Blöbaum, S., & Margellos, K. (2022). Verification of safety critical control policies using kernel methods. 1870–1875.","ama":"Vertovec N, Ober-Blöbaum S, Margellos K. Verification of safety critical control policies using kernel methods. In: ; 2022:1870-1875."},"year":"2022","type":"conference","page":"1870-1875","user_id":"15694","title":"Verification of safety critical control policies using kernel methods","ddc":["510"],"abstract":[{"text":"Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.","lang":"eng"}],"has_accepted_license":"1","status":"public","date_created":"2022-03-31T11:14:13Z","author":[{"id":"93930","last_name":"Vertovec","full_name":"Vertovec, Nikolaus","first_name":"Nikolaus"},{"id":"16494","last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina","first_name":"Sina"},{"last_name":"Margellos","first_name":"Kostas","full_name":"Margellos, Kostas"}],"department":[{"_id":"636"}]},{"year":"2021","type":"conference","citation":{"mla":"Ober-Blöbaum, Sina, and M. Vermeeren. “Superconvergence of Galerkin Variational Integrators.” 7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC, edited by IFAC-PapersOnLine, vol. 54(19), 2021, pp. 327–33.","bibtex":"@inproceedings{Ober-Blöbaum_Vermeeren_2021, title={Superconvergence of galerkin variational integrators}, volume={54(19)}, booktitle={7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC}, author={Ober-Blöbaum, Sina and Vermeeren, M.}, editor={IFAC-PapersOnLine}, year={2021}, pages={327–333} }","chicago":"Ober-Blöbaum, Sina, and M. Vermeeren. “Superconvergence of Galerkin Variational Integrators.” In 7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC, edited by IFAC-PapersOnLine, 54(19):327–33, 2021.","ama":"Ober-Blöbaum S, Vermeeren M. Superconvergence of galerkin variational integrators. In: IFAC-PapersOnLine, ed. 7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC. Vol 54(19). ; 2021:327-333.","apa":"Ober-Blöbaum, S., & Vermeeren, M. (2021). Superconvergence of galerkin variational integrators. In IFAC-PapersOnLine (Ed.), 7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC: Vol. 54(19) (pp. 327–333).","ieee":"S. Ober-Blöbaum and M. Vermeeren, “Superconvergence of galerkin variational integrators,” in 7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC, 2021, vol. 54(19), pp. 327–333.","short":"S. Ober-Blöbaum, M. Vermeeren, in: IFAC-PapersOnLine (Ed.), 7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC, 2021, pp. 327–333."},"page":"327-333","language":[{"iso":"eng"}],"corporate_editor":["IFAC-PapersOnLine"],"_id":"29421","date_updated":"2022-01-21T13:36:53Z","volume":"54(19)","status":"public","date_created":"2022-01-18T14:27:56Z","author":[{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"},{"full_name":"Vermeeren, M.","first_name":"M.","last_name":"Vermeeren"}],"publication":"7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC","department":[{"_id":"636"}],"title":"Superconvergence of galerkin variational integrators","user_id":"15694"},{"type":"journal_article","year":"2021","citation":{"ama":"Ober-Blöbaum S, Peitz S. Explicit multiobjective model predictive control for nonlinear systems with symmetries. International Journal of Robust and Nonlinear Control. 2021;31(2):380-403. doi:10.1002/rnc.5281","apa":"Ober-Blöbaum, S., & Peitz, S. (2021). Explicit multiobjective model predictive control for nonlinear systems with symmetries. International Journal of Robust and Nonlinear Control, 31(2), 380–403. https://doi.org/10.1002/rnc.5281","chicago":"Ober-Blöbaum, Sina, and Sebastian Peitz. “Explicit Multiobjective Model Predictive Control for Nonlinear Systems with Symmetries.” International Journal of Robust and Nonlinear Control 31(2) (2021): 380–403. https://doi.org/10.1002/rnc.5281.","bibtex":"@article{Ober-Blöbaum_Peitz_2021, title={Explicit multiobjective model predictive control for nonlinear systems with symmetries}, volume={31(2)}, DOI={10.1002/rnc.5281}, journal={International Journal of Robust and Nonlinear Control}, author={Ober-Blöbaum, Sina and Peitz, Sebastian}, year={2021}, pages={380–403} }","mla":"Ober-Blöbaum, Sina, and Sebastian Peitz. “Explicit Multiobjective Model Predictive Control for Nonlinear Systems with Symmetries.” International Journal of Robust and Nonlinear Control, vol. 31(2), 2021, pp. 380–403, doi:10.1002/rnc.5281.","short":"S. Ober-Blöbaum, S. Peitz, International Journal of Robust and Nonlinear Control 31(2) (2021) 380–403.","ieee":"S. Ober-Blöbaum and S. Peitz, “Explicit multiobjective model predictive control for nonlinear systems with symmetries,” International Journal of Robust and Nonlinear Control, vol. 31(2), pp. 380–403, 2021, doi: 10.1002/rnc.5281."},"page":"380-403","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/rnc.5281"}],"doi":"10.1002/rnc.5281","oa":"1","date_updated":"2022-01-24T13:27:50Z","_id":"16294","volume":"31(2)","status":"public","project":[{"_id":"52","name":"Computing Resources Provided by the Paderborn Center for Parallel Computing"}],"date_created":"2020-03-13T12:44:36Z","author":[{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"},{"orcid":"https://orcid.org/0000-0002-3389-793X","full_name":"Peitz, Sebastian","first_name":"Sebastian","id":"47427","last_name":"Peitz"}],"department":[{"_id":"101"}],"publication":"International Journal of Robust and Nonlinear Control","title":"Explicit multiobjective model predictive control for nonlinear systems with symmetries","user_id":"15694","abstract":[{"text":"Model predictive control is a prominent approach to construct a feedback\r\ncontrol loop for dynamical systems. Due to real-time constraints, the major\r\nchallenge in MPC is to solve model-based optimal control problems in a very\r\nshort amount of time. For linear-quadratic problems, Bemporad et al. have\r\nproposed an explicit formulation where the underlying optimization problems are\r\nsolved a priori in an offline phase. In this article, we present an extension\r\nof this concept in two significant ways. We consider nonlinear problems and -\r\nmore importantly - problems with multiple conflicting objective functions. In\r\nthe offline phase, we build a library of Pareto optimal solutions from which we\r\nthen obtain a valid compromise solution in the online phase according to a\r\ndecision maker's preference. Since the standard multi-parametric programming\r\napproach is no longer valid in this situation, we instead use interpolation\r\nbetween different entries of the library. To reduce the number of problems that\r\nhave to be solved in the offline phase, we exploit symmetries in the dynamical\r\nsystem and the corresponding multiobjective optimal control problem. The\r\nresults are verified using two different examples from autonomous driving.","lang":"eng"}]},{"user_id":"15694","title":"A multirate variational approach to Nonlinear MPC","publisher":"European Control Conference (ECC), IEEE","author":[{"last_name":"Lishkova","full_name":"Lishkova, Y.","first_name":"Y."},{"last_name":"Cannon","full_name":"Cannon, M.","first_name":"M."},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"}],"date_created":"2023-09-21T07:14:57Z","status":"public","date_updated":"2023-09-21T07:27:30Z","_id":"47146","language":[{"iso":"eng"}],"citation":{"short":"Y. Lishkova, M. Cannon, S. Ober-Blöbaum, in: European Control Conference (ECC), IEEE, 2021.","ieee":"Y. Lishkova, M. Cannon, and S. Ober-Blöbaum, “A multirate variational approach to Nonlinear MPC,” 2021.","ama":"Lishkova Y, Cannon M, Ober-Blöbaum S. A multirate variational approach to Nonlinear MPC. In: European Control Conference (ECC), IEEE; 2021.","apa":"Lishkova, Y., Cannon, M., & Ober-Blöbaum, S. (2021). A multirate variational approach to Nonlinear MPC.","chicago":"Lishkova, Y., M. Cannon, and Sina Ober-Blöbaum. “A Multirate Variational Approach to Nonlinear MPC.” European Control Conference (ECC), IEEE, 2021.","bibtex":"@inproceedings{Lishkova_Cannon_Ober-Blöbaum_2021, title={A multirate variational approach to Nonlinear MPC}, publisher={European Control Conference (ECC), IEEE}, author={Lishkova, Y. and Cannon, M. and Ober-Blöbaum, Sina}, year={2021} }","mla":"Lishkova, Y., et al. A Multirate Variational Approach to Nonlinear MPC. European Control Conference (ECC), IEEE, 2021."},"year":"2021","type":"conference"},{"file_date_updated":"2021-07-29T09:37:49Z","keyword":["optimal control","catastrophe theory","bifurcations","variational methods","symplectic integrators"],"author":[{"orcid":"0000-0002-5940-8057","full_name":"Offen, Christian","first_name":"Christian","id":"85279","last_name":"Offen"},{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"}],"quality_controlled":"1","file":[{"file_id":"22895","creator":"coffen","file_size":3125220,"relation":"main_file","date_updated":"2021-07-29T09:37:49Z","content_type":"application/pdf","file_name":"ifacconf.pdf","date_created":"2021-07-29T09:37:49Z","access_level":"open_access"}],"volume":"54(19)","date_created":"2021-07-29T09:38:32Z","has_accepted_license":"1","status":"public","abstract":[{"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.","lang":"eng"}],"ddc":["510"],"user_id":"15694","main_file_link":[{"open_access":"1","url":"https://www.sciencedirect.com/science/article/pii/S2405896321021236"}],"page":"334-339","citation":{"short":"C. Offen, S. Ober-Blöbaum, 54(19) (2021) 334–339.","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.","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.","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","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.","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} }"},"type":"conference","year":"2021","conference":{"location":"Berlin, Germany","name":"7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, LHMNC 2021","start_date":"2021-10-11","end_date":"2021-10-13"},"_id":"22894","department":[{"_id":"636"}],"publication_identifier":{"issn":["2405-8963"]},"publication_status":"published","external_id":{"arxiv":["2107.13853"]},"title":"Bifurcation preserving discretisations of optimal control problems","related_material":{"link":[{"description":"GitHub/Zenodo","relation":"software","url":"https://doi.org/10.5281/zenodo.4562664"}]},"series_title":"IFAC-PapersOnLine","language":[{"iso":"eng"}],"date_updated":"2023-11-29T10:19:41Z","doi":"https://doi.org/10.1016/j.ifacol.2021.11.099","oa":"1"},{"page":"2896","type":"conference","citation":{"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.","short":"S. Ridderbusch, C. Offen, S. Ober-Blöbaum, P. Goulart, in: 2021 60th IEEE Conference on Decision and Control (CDC), IEEE, 2021, p. 2896.","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} }","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.","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.","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"},"year":"2021","conference":{"location":"Austin, TX, USA","start_date":"2021-12-14","name":"60th IEEE Conference on Decision and Control (CDC)","end_date":"2021-12-17"},"_id":"21572","date_created":"2021-03-30T10:27:44Z","status":"public","publication":"2021 60th IEEE Conference on Decision and Control (CDC)","publisher":"IEEE","author":[{"last_name":"Ridderbusch","first_name":"Steffen","full_name":"Ridderbusch, Steffen"},{"last_name":"Offen","id":"85279","first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian"},{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"},{"full_name":"Goulart, Paul","first_name":"Paul","last_name":"Goulart"}],"user_id":"15694","language":[{"iso":"eng"}],"doi":"10.1109/CDC45484.2021.9683426","date_updated":"2023-11-29T10:24:55Z","publication_identifier":{"eisbn":["978-1-6654-3659-5"]},"publication_status":"published","department":[{"_id":"636"}],"title":"Learning ODE Models with Qualitative Structure Using Gaussian Processes ","related_material":{"link":[{"url":"https://github.com/Crown421/StructureGPs-paper","description":"GitHub","relation":"software"}]},"external_id":{"arxiv":["2011.05364"]}},{"language":[{"iso":"eng"}],"page":"1975-1980","type":"conference","year":"2021","citation":{"ieee":"N. Vertovec, S. Ober-Blöbaum, and K. Margellos, “Multi-objective minimum time optimal control for low-thrust trajectory design,” Rotterdam, the Netherlands, pp. 1975–1980.","short":"N. Vertovec, S. Ober-Blöbaum, K. Margellos, in: n.d., pp. 1975–1980.","mla":"Vertovec, Nikolaus, et al. Multi-Objective Minimum Time Optimal Control for Low-Thrust Trajectory Design. pp. 1975–80.","bibtex":"@inproceedings{Vertovec_Ober-Blöbaum_Margellos, title={Multi-objective minimum time optimal control for low-thrust trajectory design}, author={Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}, pages={1975–1980} }","chicago":"Vertovec, Nikolaus, Sina Ober-Blöbaum, and Kostas Margellos. “Multi-Objective Minimum Time Optimal Control for Low-Thrust Trajectory Design,” 1975–80, n.d.","apa":"Vertovec, N., Ober-Blöbaum, S., & Margellos, K. (n.d.). Multi-objective minimum time optimal control for low-thrust trajectory design. 1975–1980.","ama":"Vertovec N, Ober-Blöbaum S, Margellos K. Multi-objective minimum time optimal control for low-thrust trajectory design. In: ; :1975-1980."},"conference":{"end_date":"2021-07-02","start_date":"2021-06-29","name":"2021 European Control Conference (ECC)","location":"Rotterdam, the Netherlands"},"date_updated":"2023-11-29T10:26:49Z","_id":"21592","date_created":"2021-04-03T03:00:35Z","status":"public","publication_status":"accepted","department":[{"_id":"636"}],"author":[{"last_name":"Vertovec","id":"87056","first_name":"Nikolaus","full_name":"Vertovec, Nikolaus"},{"last_name":"Ober-Blöbaum","id":"16494","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"},{"last_name":"Margellos","full_name":"Margellos, Kostas","first_name":"Kostas"}],"user_id":"15694","title":"Multi-objective minimum time optimal control for low-thrust trajectory design","external_id":{"arxiv":["2103.08813"]},"abstract":[{"text":"We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design.","lang":"eng"}]},{"series_title":"J Nonlinear Sci ","language":[{"iso":"eng"}],"year":"2021","citation":{"apa":"Jiménez, F., & Ober-Blöbaum, S. (2021). Fractional Damping Through Restricted Calculus of Variations. Nichtlineare Sci 31, 46.","ama":"Jiménez F, Ober-Blöbaum S. Fractional Damping Through Restricted Calculus of Variations. In: Nichtlineare Sci 31. Vol 46. J Nonlinear Sci . ; 2021.","chicago":"Jiménez, F., and Sina Ober-Blöbaum. “Fractional Damping Through Restricted Calculus of Variations.” In Nichtlineare Sci 31, Vol. 46. J Nonlinear Sci , 2021.","mla":"Jiménez, F., and Sina Ober-Blöbaum. “Fractional Damping Through Restricted Calculus of Variations.” Nichtlineare Sci 31, vol. 46, 2021.","bibtex":"@inproceedings{Jiménez_Ober-Blöbaum_2021, series={J Nonlinear Sci }, title={Fractional Damping Through Restricted Calculus of Variations}, volume={46}, booktitle={Nichtlineare Sci 31}, author={Jiménez, F. and Ober-Blöbaum, Sina}, year={2021}, collection={J Nonlinear Sci } }","short":"F. Jiménez, S. Ober-Blöbaum, in: Nichtlineare Sci 31, 2021.","ieee":"F. Jiménez and S. Ober-Blöbaum, “Fractional Damping Through Restricted Calculus of Variations,” in Nichtlineare Sci 31, 2021, vol. 46."},"type":"conference","intvolume":" 46","_id":"29868","date_updated":"2023-11-29T10:23:46Z","department":[{"_id":"636"}],"publication":"Nichtlineare Sci 31","author":[{"last_name":"Jiménez","first_name":"F.","full_name":"Jiménez, F."},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","id":"16494"}],"date_created":"2022-02-17T07:28:47Z","status":"public","volume":46,"user_id":"15694","title":"Fractional Damping Through Restricted Calculus of Variations"}]