Deep learning for mean field optimal transport
ESAIM. Proceedings, Tome 77 (2024), pp. 145-175
Cet article a éte moissonné depuis la source EDP Sciences
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.
Affiliations des auteurs :
Sebastian Baudelet 1 ; Brieuc Frénais 2 ; Mathieu Laurière 3 ; Amal Machtalay 4 ; Yuchen Zhu 5
@article{EP_2024_77_a7,
author = {Sebastian Baudelet and Brieuc Fr\'enais and Mathieu Lauri\`ere and Amal Machtalay and Yuchen Zhu},
title = {Deep learning for mean field optimal transport},
journal = {ESAIM. Proceedings},
pages = {145--175},
year = {2024},
volume = {77},
doi = {10.1051/proc/202477145},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202477145/}
}
TY - JOUR AU - Sebastian Baudelet AU - Brieuc Frénais AU - Mathieu Laurière AU - Amal Machtalay AU - Yuchen Zhu TI - Deep learning for mean field optimal transport JO - ESAIM. Proceedings PY - 2024 SP - 145 EP - 175 VL - 77 UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/202477145/ DO - 10.1051/proc/202477145 LA - en ID - EP_2024_77_a7 ER -
%0 Journal Article %A Sebastian Baudelet %A Brieuc Frénais %A Mathieu Laurière %A Amal Machtalay %A Yuchen Zhu %T Deep learning for mean field optimal transport %J ESAIM. Proceedings %D 2024 %P 145-175 %V 77 %U http://geodesic.mathdoc.fr/articles/10.1051/proc/202477145/ %R 10.1051/proc/202477145 %G en %F EP_2024_77_a7
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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