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

Lars Ruthotto1,2, Stanley J Osher3, Wuchen Li4

  • 1Department of Mathematics, Emory University, Atlanta, GA 30322; sjo@math.ucla.edu lruthotto@emory.edu.

Proceedings of the National Academy of Sciences of the United States of America
|April 11, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a machine learning framework to solve high-dimensional mean field games (MFG) and mean field control (MFC) problems. The new method overcomes the curse of dimensionality, enabling solutions for complex multiagent systems.

Keywords:
Hamilton-Jacobi-Bellman equationsmachine learningmean field controlmean field gamesoptimal transport

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Area of Science:

  • Multiagent systems and game theory
  • Numerical analysis and scientific computing
  • Machine learning applications

Background:

  • Mean field games (MFG) and mean field control (MFC) are essential for analyzing large populations of interacting agents in fields like economics and crowd motion.
  • Traditional numerical methods face a curse of dimensionality due to spatial discretization, limiting their application to high-dimensional problems.
  • Existing methods struggle with the computational complexity of large-scale multiagent systems.

Purpose of the Study:

  • To develop a flexible machine learning framework for the numerical solution of potential MFG and MFC models.
  • To overcome the limitations of existing numerical methods in solving high-dimensional MFG and MFC problems.
  • To enable the application of MFG and MFC models to previously intractable problems.

Main Methods:

  • Combines Lagrangian and Eulerian viewpoints with recent machine learning advances.
  • Utilizes a Lagrangian formulation while enforcing the derived Eulerian Hamilton-Jacobi-Bellman (HJB) equation.
  • Employs a tailored neural network parameterization to avoid spatial discretization, addressing the curse of dimensionality.

Main Results:

  • Successfully approximated solutions for 100-dimensional optimal transport and crowd motion problems.
  • Demonstrated the framework's capability on a standard workstation, showcasing computational efficiency.
  • Validated results using a traditional Eulerian solver in two dimensions.

Conclusions:

  • The proposed machine learning framework offers a flexible and efficient approach to solving high-dimensional MFG and MFC problems.
  • This methodology significantly expands the scope of applications for mean field models beyond current limitations.
  • The ability to handle high-dimensional instances opens new avenues for research and practical implementation in various scientific and engineering domains.