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. We call such problems mean field optimal transport problems 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.

翻译：平均场控制问题已被引入用于研究大规模策略型群体中的社会最优。其核心思想是考虑无限人口规模，并通过平均场近似简化分析。此类问题也可视为McKean-Vlasov动力学的最优控制问题，在经济学、金融学、社会科学及工程学等诸多领域均有应用。通常，智能体的目标是使总成本最小化，该总成本由运行成本积分与终端成本组成。本文研究无终端成本但终端分布预设的平均场控制问题，我们将其称为平均场最优输运问题——当动力学或运行成本函数中存在平均场相互作用时，此类问题可视为经典最优输运的推广。我们提出三种基于神经网络的方法：第一种直接学习最优控制，第二种求解表征解的FBSDE耦合偏微分方程组，第三种采用原对偶方法。通过两类实例的数值实验对所提方法进行了验证。