{"publication_status":"submitted","external_id":{"arxiv":["2407.07642"]},"_id":"55159","status":"public","page":"28","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."}],"related_material":{"link":[{"description":"GitHub","relation":"software","url":"https://github.com/Christian-Offen/Lagrangian_GP_PDE"}]},"keyword":["System identification","inverse problem of variational calculus","Gaussian process","Lagrangian learning","physics informed machine learning","geometry aware learning"],"user_id":"85279","file_date_updated":"2024-07-10T13:39:32Z","oa":"1","title":"Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification","language":[{"iso":"eng"}],"date_updated":"2024-08-12T13:43:32Z","author":[{"first_name":"Christian","id":"85279","last_name":"Offen","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian"}],"has_accepted_license":"1","type":"preprint","file":[{"file_name":"L_Collocation.pdf","access_level":"open_access","date_created":"2024-07-10T13:39:32Z","file_id":"55160","date_updated":"2024-07-10T13:39:32Z","description":"We introduce a method based on Gaussian process regression to identify discrete\nvariational 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.\nWe provide a rigorous convergence statement of the method.\nThe proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus.\nMoreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory.\nThis is illustrated on the example of the discrete wave equation and Schrödinger equation.\nThe 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.","relation":"main_file","content_type":"application/pdf","file_size":4569314,"title":"Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification","creator":"coffen"}],"year":"2024","department":[{"_id":"636"}],"ddc":["510"],"project":[{"_id":"52","name":"PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing"}],"date_created":"2024-07-10T13:43:50Z","citation":{"chicago":"Offen, Christian. “Machine Learning of Discrete Field Theories with Guaranteed Convergence and Uncertainty Quantification,” n.d.","bibtex":"@article{Offen, title={Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification}, author={Offen, Christian} }","mla":"Offen, Christian. <i>Machine Learning of Discrete Field Theories with Guaranteed Convergence and Uncertainty Quantification</i>.","short":"C. Offen, (n.d.).","ieee":"C. Offen, “Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification.” .","apa":"Offen, C. (n.d.). <i>Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification</i>.","ama":"Offen C. Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification."}}