A Machine Learning Framework for Solving High-Dimensional Mean Field\n Game and Mean Field Control Problems

Abstract

Mean field games (MFG) and mean field control (MFC) are critical classes of\nmulti-agent models for efficient analysis of massive populations of interacting\nagents. Their areas of application span topics in economics, finance, game\ntheory, industrial engineering, crowd motion, and more. In this paper, we\nprovide a flexible machine learning framework for the numerical solution of\npotential MFG and MFC models. State-of-the-art numerical methods for solving\nsuch problems utilize spatial discretization that leads to a\ncurse-of-dimensionality. We approximately solve high-dimensional problems by\ncombining Lagrangian and Eulerian viewpoints and leveraging recent advances\nfrom machine learning. More precisely, we work with a Lagrangian formulation of\nthe problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation\nthat is derived from the Eulerian formulation. Finally, a tailored neural\nnetwork parameterization of the MFG/MFC solution helps us avoid any spatial\ndiscretization. Our numerical results include the approximate solution of\n100-dimensional instances of optimal transport and crowd motion problems on a\nstandard work station and a validation using an Eulerian solver in two\ndimensions. These results open the door to much-anticipated applications of MFG\nand MFC models that were beyond reach with existing numerical methods.\n