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 a 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.