Geodesic
Deep learning for mean field optimal transport
Sebastian Baudelet1 ; Brieuc Frénais2 ; Mathieu Laurière3 ; Amal Machtalay4 ; Yuchen Zhu5
1Université Côte d’Azur, 28 Avenue de Valrose, 06103 Nice, France
2IRMA UMR 7501, Université de Strasbourg, 7 Rue René Descartes, 67000 Strasbourg, France
3NYU Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning; NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China
4Mohammed VI Polytechnic University. Lot 660, Hay Moulay Rachid Ben Guerir, 43150, Morocco
5Georgia Institute of Technology, 686 Cherry Street NW, Atlanta, 30332, USA
ESAIM. Proceedings, Tome 77 (2024), pp. 145-175
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Mean field control (MFC) problems have been introduced to study social optima in very large populations of strategic agents. The main idea is to consider an infinite population and to simplify the analysis by using a mean field approximation. These problems can also be viewed as optimal control problems for McKean-Vlasov dynamics. They have found applications in a wide range of fields, from economics and finance to social sciences and engineering. Usually, the goal for the agents is to minimize a total cost which consists in the integral of a running cost plus a terminal cost. In this work, we consider MFC problems in which there is no terminal cost but, instead, the terminal distribution is prescribed as in optimal transport problem. By analogy with MFC, we call such problems mean field optimal transport problems (or MFOT for short) since they can be viewed as a generalization of classical optimal transport problems when mean field interactions occur in the dynamics or the running cost function. We propose three numerical methods based on neural networks. The first one is based on directly learning an optimal control. The second one amounts to solve a forward-backward PDE system characterizing the solution. The third one relies on a primal-dual approach. We illustrate these methods with numerical experiments conducted on two families of examples.
DOI : 10.1051/proc/202477145

Sebastian Baudelet  1   ; Brieuc Frénais  2   ; Mathieu Laurière  3   ; Amal Machtalay  4   ; Yuchen Zhu  5

1 Université Côte d’Azur, 28 Avenue de Valrose, 06103 Nice, France
2 IRMA UMR 7501, Université de Strasbourg, 7 Rue René Descartes, 67000 Strasbourg, France
3 NYU Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning; NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China
4 Mohammed VI Polytechnic University. Lot 660, Hay Moulay Rachid Ben Guerir, 43150, Morocco
5 Georgia Institute of Technology, 686 Cherry Street NW, Atlanta, 30332, USA 
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     title = {Deep learning for mean field optimal transport},
     journal = {ESAIM. Proceedings},
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Sebastian Baudelet; Brieuc Frénais; Mathieu Laurière; Amal Machtalay; Yuchen Zhu. Deep learning for mean field optimal transport. ESAIM. Proceedings, Tome 77 (2024), pp. 145-175. doi: 10.1051/proc/202477145

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