{"id":{"repo_id":"umn","oai_identifier":"oai:conservancy.umn.edu:11299/269198"},"canonical_url":"https://search.dev.ndltd.org/etd/umn/oai:conservancy.umn.edu:11299/269198","repository":{"repo_id":"umn","name":"University of Minnesota","base_url":"https://conservancy.umn.edu/server/oai/request"},"display":{"title":"Introducing the Picard Method for Approximating Solutions of Differential Equations with Neural Networks","abstract":"For decades, differential equations have been used to model various problems in the natural sciences and engineering. However, ordinary differential equations (ODEs) are usually not analytically solvable, so many numerical approaches have been developed to produce approximate solutions. More recently, it has been proposed that neural networks can learn solutions of ODEs and thus provide faster and more accurate numerical approximations. Here, we propose a novel approach of having neural networks learn solutions to ODEs via the Picard formulation. We show, through examples, that this approach produces approximations that are at least as reliable as earlier approaches.","abstract_html":"For decades, differential equations have been used to model various problems in the natural sciences and engineering. However, ordinary differential equations (ODEs) are usually not analytically solvable, so many numerical approaches have been developed to produce approximate solutions. More recently, it has been proposed that neural networks can learn solutions of ODEs and thus provide faster and more accurate numerical approximations. Here, we propose a novel approach of having neural networks learn solutions to ODEs via the Picard formulation. 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May 2024. Major: Mathematics. Advisors: Richard McGehee, Jeff Calder. 1 computer file (PDF); vi, 51 pages."]},{"key":"dc:description.abstract","label":"Abstract","values":["For decades, differential equations have been used to model various problems in the natural sciences and engineering. However, ordinary differential equations (ODEs) are usually not analytically solvable, so many numerical approaches have been developed to produce approximate solutions. More recently, it has been proposed that neural networks can learn solutions of ODEs and thus provide faster and more accurate numerical approximations. Here, we propose a novel approach of having neural networks learn solutions to ODEs via the Picard formulation. We show, through examples, that this approach produces approximations that are at least as reliable as earlier approaches."]},{"key":"dc:title","label":"Title","values":["Introducing the Picard Method for Approximating Solutions of Differential Equations with Neural Networks"]}]}],"canonical_facts":{"dc:creator":["Frazier, Ty"],"dc:date.accessioned":["2025-01-07T16:11:57Z"],"dc:date.available":["2025-01-07T16:11:57Z"],"dc:date.issued":["2024-05"],"dc:description":["University of Minnesota Ph.D. dissertation. May 2024. Major: Mathematics. Advisors: Richard McGehee, Jeff Calder. 1 computer file (PDF); vi, 51 pages."],"dc:description.abstract":["For decades, differential equations have been used to model various problems in the natural sciences and engineering. However, ordinary differential equations (ODEs) are usually not analytically solvable, so many numerical approaches have been developed to produce approximate solutions. More recently, it has been proposed that neural networks can learn solutions of ODEs and thus provide faster and more accurate numerical approximations. Here, we propose a novel approach of having neural networks learn solutions to ODEs via the Picard formulation. We show, through examples, that this approach produces approximations that are at least as reliable as earlier approaches."],"dc:identifier.uri":["https://hdl.handle.net/11299/269198"],"dc:language.iso":["en"],"dc:subject":["Dynamical Systems","Neural Networks","Ordinary Differential Equations"],"dc:title":["Introducing the Picard Method for Approximating Solutions of Differential Equations with Neural Networks"],"dc:type":["Thesis or Dissertation"]},"updated_at":"2026-07-24T05:19:48Z"}