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:在大量战略剂中,为研究社会选择,引入了中流场控制(MFC)问题,在大量战略剂中研究社会选择(MFC)问题,主要思想是考虑无限人口,通过使用中流场近似来简化分析。这些问题也可以被视为McKan-Vlasov动态的最佳控制问题。这些问题在从经济和金融到社会科学和工程等广泛领域都发现了应用。通常,代理机构的目标是最大限度地减少总成本,这包括运行成本和终端成本。在这项工作中,我们考虑到MFC问题,在这些问题上没有终端成本,而是规定了终端分布。我们称之为外地最佳运输问题,因为当动态或运行成本功能中出现典型的最佳运输问题时,这些问题可以被看作是典型的实地互动的典型问题。我们提出基于神经网络的三种数字方法。第一个方法基于直接学习最佳控制。第二个是解决前向后方PDE系统特征的解决方案。第三个方法依靠原始方法。我们用两个家庭进行的数字实验来说明这些方法。</s>