{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T14:49:33Z","timestamp":1766414973786,"version":"3.37.3"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,10,25]],"date-time":"2022-10-25T00:00:00Z","timestamp":1666656000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,10,25]],"date-time":"2022-10-25T00:00:00Z","timestamp":1666656000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/V008331\/1"],"award-info":[{"award-number":["EP\/V008331\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We discuss the numerical solution to a class of continuous time finite state mean field games. We apply the deep neural network (DNN) approach to solving the fully coupled forward and backward ordinary differential equation system that characterizes the equilibrium value function and probability measure of the finite state mean field game. We prove that the error between the true solution and the approximate solution is linear to the square root of DNN loss function. We give an example of applying the DNN method to solve the optimal market making problem with terminal rank-based trading volume reward.<\/jats:p>","DOI":"10.1007\/s13235-022-00477-5","type":"journal-article","created":{"date-parts":[[2022,10,25]],"date-time":"2022-10-25T10:02:46Z","timestamp":1666692166000},"page":"859-896","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Deep Neural Network Solution for Finite State Mean Field Game with Error Estimation"],"prefix":"10.1007","volume":"13","author":[{"given":"Jialiang","family":"Luo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5882-2088","authenticated-orcid":false,"given":"Harry","family":"Zheng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,10,25]]},"reference":[{"key":"477_CR1","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1137\/100790069","volume":"50","author":"Y Achdou","year":"2012","unstructured":"Achdou Y, Camilli F, Capuzzo-Dolcetta I (2012) Mean field games: numerical methods for the planning problem. SIAM J Control Optim 50:77\u2013109","journal-title":"SIAM J Control Optim"},{"key":"477_CR2","doi-asserted-by":"publisher","first-page":"1136","DOI":"10.1137\/090758477","volume":"48","author":"Y Achdou","year":"2010","unstructured":"Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48:1136\u20131162","journal-title":"SIAM J Numer Anal"},{"key":"477_CR3","unstructured":"Cardaliaguet P, Delarue F, Lasry JM, Lions PL (2015) The master equation and the convergence problem in mean field games. arXiv:1509.02505"},{"key":"477_CR4","doi-asserted-by":"publisher","first-page":"2705","DOI":"10.1137\/120883499","volume":"51","author":"R Carmona","year":"2013","unstructured":"Carmona R, Delarue F (2013) Probabilistic analysis of mean-field games. SIAM J Control Optim 51:2705\u20132734","journal-title":"SIAM J Control Optim"},{"key":"477_CR5","doi-asserted-by":"crossref","unstructured":"Carmona R, Lauri\u00e8re M (2021) Deep learning for mean field games and mean field control with applications to finance. arXiv:2107.04568","DOI":"10.1090\/psapm\/078\/06"},{"key":"477_CR6","doi-asserted-by":"crossref","unstructured":"Carmona R, Wang P (2016) Finite state mean field games with major and minor players. arXiv:1610.05408","DOI":"10.1214\/15-AAP1125"},{"key":"477_CR7","unstructured":"Carmona R, Wang P (2018) A probabilistic approach to extended finite state mean field games. arXiv:1808.07635"},{"key":"477_CR8","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/s00245-018-9488-7","volume":"81","author":"A Cecchin","year":"2020","unstructured":"Cecchin A, Fischer M (2020) Probabilistic approach to finite state mean field games. Appl Math Optim 81:253\u2013300","journal-title":"Appl Math Optim"},{"key":"477_CR9","doi-asserted-by":"publisher","first-page":"4510","DOI":"10.1016\/j.spa.2018.12.002","volume":"129","author":"A Cecchin","year":"2019","unstructured":"Cecchin A, Pelino G (2019) Convergence, fluctuations and large deviations for finite state mean field games via the master equation. Stochast Process Appl 129:4510\u20134555","journal-title":"Stochast Process Appl"},{"key":"477_CR10","doi-asserted-by":"crossref","unstructured":"El\u00a0Euch O, Mastrolia T, Rosenbaum M, Touzi N (2018) Optimal make-take fees for market making regulation. SSRN 3174933","DOI":"10.2139\/ssrn.3174933"},{"key":"477_CR11","doi-asserted-by":"publisher","first-page":"11","DOI":"10.3389\/fams.2020.00011","volume":"6","author":"JP Fouque","year":"2020","unstructured":"Fouque JP, Zhang Z (2020) Deep learning methods for mean field control problems with delay. Front Appl Math Stat 6:11","journal-title":"Front Appl Math Stat"},{"key":"477_CR12","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s00245-013-9202-8","volume":"68","author":"D Gomes","year":"2013","unstructured":"Gomes D, Mohr J, Souza R (2013) Continuous time finite state mean field games. Appl Math Optim 68:99\u2013143","journal-title":"Appl Math Optim"},{"key":"477_CR13","unstructured":"Gomes D, Saude J (2017) Monotone numerical methods for finite-state mean-field games. arXiv:1705.00174"},{"key":"477_CR14","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1016\/j.matpur.2009.04.008","volume":"92","author":"O Gu\u00e9ant","year":"2009","unstructured":"Gu\u00e9ant O (2009) A reference case for mean field games models. J math\u00e9matiques pures et appliqu\u00e9es 92:276\u2013294","journal-title":"J math\u00e9matiques pures et appliqu\u00e9es"},{"key":"477_CR15","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1080\/1350486X.2017.1342552","volume":"24","author":"O Gu\u00e9ant","year":"2017","unstructured":"Gu\u00e9ant O (2017) Optimal market making. Appl Math Finance 24:112\u2013154","journal-title":"Appl Math Finance"},{"key":"477_CR16","doi-asserted-by":"crossref","unstructured":"Gu\u00e9ant O, Lasry JM, Lions PL (2011) Mean field games and applications. In: Paris-Princeton lectures on mathematical finance 2010. Springer, pp 205\u2013266","DOI":"10.1007\/978-3-642-14660-2_3"},{"key":"477_CR17","doi-asserted-by":"publisher","first-page":"8505","DOI":"10.1073\/pnas.1718942115","volume":"115","author":"J Han","year":"2018","unstructured":"Han J, Jentzen A (2018) Solving high-dimensional partial differential equations using deep learning. Proc Natl Acad Sci 115:8505\u20138510","journal-title":"Proc Natl Acad Sci"},{"key":"477_CR18","doi-asserted-by":"publisher","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","volume":"6","author":"M Huang","year":"2006","unstructured":"Huang M, Malham\u00e9 R, Caines P (2006) Large population stochastic dynamic games: closed-loop Mckean-Vlasov systems and the nash certainty equivalence principle. Commun Inf Syst 6:221\u2013252","journal-title":"Commun Inf Syst"},{"key":"477_CR19","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","volume":"9","author":"I Lagaris","year":"1998","unstructured":"Lagaris I, Likas A, Fotiadis D (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9:987\u20131000","journal-title":"IEEE Trans Neural Netw"},{"key":"477_CR20","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1109\/72.870037","volume":"11","author":"I Lagaris","year":"2000","unstructured":"Lagaris I, Likas A, Papageorgiou D (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11:1041\u20131049","journal-title":"IEEE Trans Neural Netw"},{"key":"477_CR21","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s11537-007-0657-8","volume":"2","author":"JM Lasry","year":"2007","unstructured":"Lasry JM, Lions PL (2007) Mean field games. Japan J Math 2:229\u2013260","journal-title":"Japan J Math"},{"key":"477_CR22","unstructured":"Lasry JM, Lions PL, Gu\u00e9ant O (2008) Application of mean field games to growth theory. hal:00348376"},{"key":"477_CR23","doi-asserted-by":"crossref","unstructured":"Lauriere M (2021) Numerical methods for mean field games and mean field type control. arXiv:2106.06231","DOI":"10.1090\/psapm\/078\/06"},{"key":"477_CR24","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1016\/0021-9991(90)90007-N","volume":"91","author":"H Lee","year":"1990","unstructured":"Lee H, Kang I (1990) Neural algorithm for solving differential equations. J Comput Phys 91:110\u2013131","journal-title":"J Comput Phys"},{"key":"477_CR25","unstructured":"Li J, Yue J, Zhang W, Duan W (2020) The deep learning Galerkin method for the general stokes equations. arXiv:2009.11701"},{"key":"477_CR26","first-page":"427","volume":"18","author":"J Li","year":"2021","unstructured":"Li J, Zhang W, Yue J (2021) A deep learning Galerkin method for the second-order linear elliptic equations. Int J Numer Anal Model 18:427\u2013441","journal-title":"Int J Numer Anal Model"},{"key":"477_CR27","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.amc.2006.05.068","volume":"183","author":"A Malek","year":"2006","unstructured":"Malek A, Beidokhti R (2006) Numerical solution for high order differential equations using a hybrid neural network-optimization method. Appl Math Comput 183:260\u2013271","journal-title":"Appl Math Comput"},{"key":"477_CR28","doi-asserted-by":"crossref","unstructured":"Mishra S, Molinaro R (2021) Estimates on the generalization error of physics-informed neural networks for approximating pdes. arXiv:2006.16144","DOI":"10.1093\/imanum\/drab032"},{"key":"477_CR29","doi-asserted-by":"publisher","first-page":"9183","DOI":"10.1073\/pnas.1922204117","volume":"117","author":"L Ruthotto","year":"2020","unstructured":"Ruthotto L, Osher SJ, Li W, Nurbekyan L, Fung SW (2020) A machine learning framework for solving high-dimensional mean field game and mean field control problems. Proc Natl Acad Sci 117:9183\u20139193","journal-title":"Proc Natl Acad Sci"},{"key":"477_CR30","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","volume":"375","author":"J Sirignano","year":"2018","unstructured":"Sirignano J, Spiliopoulos K (2018) DGM: a deep learning algorithm for solving partial differential equations. J Comput Phys 375:1339\u20131364","journal-title":"J Comput Phys"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-022-00477-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-022-00477-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-022-00477-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T17:31:57Z","timestamp":1691688717000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-022-00477-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,25]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["477"],"URL":"https:\/\/doi.org\/10.1007\/s13235-022-00477-5","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"type":"print","value":"2153-0785"},{"type":"electronic","value":"2153-0793"}],"subject":[],"published":{"date-parts":[[2022,10,25]]},"assertion":[{"value":"27 September 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}