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An important application of this method is to solve the path planning problem of multiagent vehicles in real time. The new method extends our previous SympOCNet framework by introducing a time-dependent symplectic network into the architecture. In addition, we propose a more general latent representation, which greatly improves model expressivity based on the universal approximation theorem. We demonstrate the efficacy of TSympOCNet in path planning problems with obstacle and collision avoidance, including systems with Newtonian dynamics and nonconvex environments, up to dimension 512. Our method shows significant promise in handling efficiently both complex dynamics and constraints.<\/jats:p>","DOI":"10.1137\/24m1682993","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T07:08:17Z","timestamp":1752131297000},"page":"C769-C794","source":"Crossref","is-referenced-by-count":0,"title":["A Time-Dependent Symplectic Network for Nonconvex Path Planning Problems with Linear and Nonlinear Dynamics"],"prefix":"10.1137","volume":"47","author":[{"given":"Zhen","family":"Zhang","sequence":"first","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chenye","family":"Wang","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shanqing","family":"Liu","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0483-7919","authenticated-orcid":true,"given":"J\u00e9r\u00f4me","family":"Darbon","sequence":"additional","affiliation":[{"name":"Corresponding author.\u00a0Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,7,10]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/060655286"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ifacol.2023.10.628"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/23M1563657"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/18M1203900"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/15M1039596"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-019-09767-9"},{"key":"ref7","doi-asserted-by":"crossref","unstructured":"S. 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