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

Machine Learning;Numerical Analysis;Numerical Analysis;Optimization and Control;Machine Learning