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

TL;DR

This paper introduces a machine learning framework that efficiently solves high-dimensional mean field game and control problems by combining Lagrangian and Eulerian approaches with neural networks, overcoming the curse of dimensionality.

Contribution

The paper presents a novel neural network-based method that avoids spatial discretization for high-dimensional MFG and MFC problems, enabling solutions in 100 dimensions.

Findings

Successfully solved 100-dimensional problems on standard hardware.

Validated results with a 2D Eulerian solver.

Demonstrated applicability to optimal transport and crowd motion.

Abstract

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.