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Author Notes:

sjo@math.ucla.edu; lruthotto@emory.edu

Author contributions: L.R., S.J.O., W.L., L.N., and S.W.F. designed research; L.R., L.N., and S.W.F. performed research; and L.R., S.J.O., W.L., L.N., and S.W.F. wrote the paper.

We thank Derek Onken for fruitful discussions and his help with the Julia implementation; and Wonjun Lee and Siting Liu for their assistance with the implementation of the Eulerian solver for the 2D problem instances and their suggestions that helped improve the manuscript.

The authors declare no competing interest.

Subjects:

Research Funding:

L.R. is supported by NSF Grant DMS-1751636. This research was performed while L.R. was visiting the Institute for Pure and Applied Mathematics, Los Angeles, CA, which is supported by NSF Grant DMS-1440415. S.J.O., W.L., L.N., and S.W.F. receive support from Air Force Office of Science Research Grants MURI-FA9550-18-1-0502 and FA9550-18-1-0167, and Office of Naval Research Grant N00014-18-1-2527.

Keywords:

  • Science & Technology
  • Multidisciplinary Sciences
  • Science & Technology - Other Topics
  • mean field games
  • mean field control
  • machine learning
  • optimal transport
  • Hamilton-Jacobi-Bellman equations
  • MASS
  • CONGESTION
  • ALGORITHM
  • EQUATIONS
  • AVERSION
  • LOADS

A machine learning framework for solving high-dimensional mean field game and mean field control problems

Tools:

Journal Title:

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA

Volume:

Volume 117, Number 17

Publisher:

, Pages 9183-9193

Type of Work:

Article | Final Publisher PDF

Abstract:

Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the 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 are beyond reach with existing numerical methods.

Copyright information:

Published under the PNAS license

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