{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:30:41Z","timestamp":1787337041010,"version":"build-2736575974"},"reference-count":94,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","funder":[{"name":"OSD\/AFOSR MURI","award":["FA9550-20-1-0358"],"award-info":[{"award-number":["FA9550-20-1-0358"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:p>Solving high-dimensional optimal control problems in real-time is an important but challenging problem, with applications to multiagent path planning problems, which have drawn increased attention given the growing popularity of drones in recent years. In this paper, we propose a novel neural network method called SympOCnet that applies the symplectic network to solve high-dimensional optimal control problems with state constraints. We present several numerical results on path planning problems in two- and three-dimensional spaces. Specifically, we demonstrate that our SympOCnet can solve a problem with more than 500 dimensions in 1.5 hours on a single GPU, which shows the effectiveness and efficiency of SympOCnet. The proposed method is scalable and has the potential to solve truly high-dimensional path planning problems in real-time.<\/jats:p>","DOI":"10.1137\/22m1472206","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T11:27:35Z","timestamp":1668684455000},"page":"B1341-B1368","source":"Crossref","is-referenced-by-count":13,"title":["SympOCnet: Solving Optimal Control Problems with Applications to High-Dimensional Multiagent Path Planning Problems"],"prefix":"10.1137","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7467-601X","authenticated-orcid":true,"given":"Tingwei","family":"Meng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhen","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0483-7919","authenticated-orcid":true,"given":"Jerome","family":"Darbon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,11,17]]},"reference":[{"key":"atypb1","volume-title":"Handbook of Linear Algebra","author":"Akian M.","year":"2007"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/060655286"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/18M1203900"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/15M1039596"},{"key":"atypb5","volume-title":"Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Numerical Applications, preprint, https:\/\/arxiv.org\/abs\/1812.05916","author":"Bachouch A.","year":"2018"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA48506.2021.9561949"},{"key":"atypb7","first-page":"4755","volume-title":"Birkh\u00e4user Boston","author":"Bardi M.","year":"1997"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1515\/9781400874668"},{"key":"atypb9","volume-title":"Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics","author":"Bertsekas D. P.","year":"1982"},{"key":"atypb10","volume-title":"Reinforcement Learning and Optimal Control","author":"Bertsekas D. P.","year":"2019"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2022.01.016"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-012-9648-x"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/ECC.2015.7331044"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.2514\/1.G000774"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2015.7402390"},{"key":"atypb16","volume-title":"Hopf-type Representation Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems, preprint, https:\/\/arxiv.org\/abs\/2110.02541","author":"Chen P.","year":"2021"},{"key":"atypb17","volume-title":"Lax-Oleinik-type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems, preprint, https:\/\/arxiv.org\/abs\/2109.14849","author":"Chen P.","year":"2021"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/0728030"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1109\/INFCOMW.2019.8845287"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/130944163"},{"key":"atypb21","volume-title":"Neural Network Architectures Using Min Plus Algebra for Solving Certain High Dimensional Optimal Control Problems and Hamilton-Jacobi PDEs, preprint, https:\/\/arxiv.org\/abs\/2105.03336","author":"Darbon J.","year":"2021"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s40687-020-00215-6"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/19M1266332"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109907"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-022-01073-3"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1186\/s40687-016-0068-7"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/978-4-431-54475-3_12"},{"key":"atypb28","first-page":"319","volume-title":"Proceedings of the IEEE-RAS International Conference on Humanoid Robots","author":"Denk J.","year":"2001"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2006.377184"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/19M1305136"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974072.51"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA.2013.6631013"},{"key":"atypb33","volume-title":"An Introduction to Mathematical Optimal Control Theory Version 0.2, lecture notes","author":"Evans L. C.","year":"1983"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.2514\/2.4709"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.2514\/2.4862"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1002\/rob.21546"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1109\/HUMANOIDS.2014.7041347"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998332433"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1109\/IROS.2007.4399327"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-016-0240-7"},{"key":"atypb41","unstructured":"D. Garg,\n                      Advances in Global Pseudospectral Methods for Optimal Control\n                      , Ph.D. thesis, University of Florida, Gainesville, FL, 2011."},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2011.6161386"},{"key":"atypb43","volume-title":"Advances in Neural Information Processing Systems 32, H. Wallach et al., eds., Curran Associates","author":"Greydanus S.","year":"2019"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.4171\/owr\/2006\/14"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1718942115"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1109\/ICRA.2016.7487459"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2014.7040310"},{"key":"atypb48","volume-title":"Some Machine Learning Schemes for High-Dimensional Nonlinear PDEs, preprint, https:\/\/arxiv.org\/abs\/1902.01599v1","author":"Hur\u00e9 C.","year":"2019"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1137\/20M1316640"},{"key":"atypb50","volume-title":"Using Neural Networks to Compute Approximate and Guaranteed Feasible Hamilton-Jacobi-Bellman PDE Solutions, preprint, https:\/\/arxiv.org\/abs\/1611.03158","author":"Jiang F.","year":"2016"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2018.01.002"},{"key":"atypb52","volume-title":"Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks, preprint, https:\/\/arxiv.org\/abs\/2012.03133","author":"Jin P.","year":"2020"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2020.08.017"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1137\/19M1262139"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1137\/17M1116635"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-017-9910-0"},{"key":"atypb57","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(00)00045-5"},{"key":"atypb58","volume-title":"Proceedings of the 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, 2015, Conference Track Proceedings","author":"Kingma D. P.","year":"2015"},{"key":"atypb59","doi-asserted-by":"publisher","DOI":"10.1109\/IROS45743.2020.9340864"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1007\/s10514-015-9479-3"},{"key":"atypb61","doi-asserted-by":"publisher","DOI":"10.1137\/030600485"},{"key":"atypb62","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109710"},{"key":"atypb63","doi-asserted-by":"publisher","DOI":"10.1109\/LCSYS.2020.3009933"},{"key":"atypb64","volume-title":"A Computationally Efficient Hamilton-Jacobi-based Formula for State-Constrained Optimal Control Problems, preprint, https:\/\/arxiv.org\/abs\/2106.13440","author":"Lee D.","year":"2021"},{"key":"atypb65","volume-title":"Robot Manipulator Control: Theory and Practice","author":"Lewis F.","year":"2004","edition":"2"},{"key":"atypb66","first-page":"500","volume-title":"PMLR","author":"Li A.","year":"2020"},{"key":"atypb67","doi-asserted-by":"publisher","DOI":"10.1109\/70.660845"},{"key":"atypb68","doi-asserted-by":"publisher","DOI":"10.1038\/s42256-021-00302-5"},{"key":"atypb69","doi-asserted-by":"publisher","DOI":"10.1109\/AERO50100.2021.9438511"},{"key":"atypb70","doi-asserted-by":"publisher","DOI":"10.1137\/040610830"},{"key":"atypb71","volume-title":"Systems Control Found. Appl.","author":"McEneaney W. M.","year":"2006"},{"key":"atypb72","doi-asserted-by":"publisher","DOI":"10.1109\/ACC.2008.4587234"},{"key":"atypb73","doi-asserted-by":"publisher","DOI":"10.1137\/070681934"},{"key":"atypb74","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.109020"},{"key":"atypb75","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4073-8"},{"key":"atypb76","doi-asserted-by":"publisher","DOI":"10.1137\/19M1288802"},{"key":"atypb77","doi-asserted-by":"publisher","DOI":"10.1109\/LCSYS.2020.3034415"},{"key":"atypb78","doi-asserted-by":"publisher","DOI":"10.1109\/CDC.2006.377358"},{"key":"atypb79","volume-title":"A Neural Network Approach for High-dimensional Optimal Control, preprint, https:\/\/arxiv.org\/abs\/2104.03270","author":"Onken D.","year":"2021"},{"key":"atypb80","doi-asserted-by":"publisher","DOI":"10.1002\/oca.2389"},{"key":"atypb81","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"atypb82","first-page":"497","volume":"135","author":"Rao A. V.","year":"2009","journal-title":"Adv. Astronautical Sci."},{"key":"atypb83","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530520500116"},{"key":"atypb84","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2018.2794582"},{"key":"atypb85","doi-asserted-by":"publisher","DOI":"10.1016\/j.arcontrol.2012.09.002"},{"key":"atypb86","volume-title":"Recursive Regression with Neural Networks: Approximating the HJI PDE Solution, preprint, https:\/\/arxiv.org\/abs\/1611.02739","author":"Royo V. R.","year":"2016"},{"key":"atypb87","doi-asserted-by":"publisher","DOI":"10.1109\/TCST.2014.2377777"},{"key":"atypb88","doi-asserted-by":"publisher","DOI":"10.1109\/TAES.2017.2767958"},{"key":"atypb89","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.08.029"},{"key":"atypb90","doi-asserted-by":"publisher","DOI":"10.1109\/TCST.2017.2709268"},{"key":"atypb91","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0710743106"},{"key":"atypb92","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-018-9509-6"},{"key":"atypb93","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2021.0207"},{"key":"atypb94","doi-asserted-by":"publisher","DOI":"10.1137\/21M1402303"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1472206","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:42:05Z","timestamp":1787334125000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1472206"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,17]]},"references-count":94,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["10.1137\/22M1472206"],"URL":"https:\/\/doi.org\/10.1137\/22m1472206","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,17]]}}}