Mean-field games (MFG) 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 data science. On example of MFGs with many applications in medical image analysis is optimal transport (OT). The OT problem for high-dimensional data sets is still challenging to solve numerically since methods that utilize space discretization are prone to the curse-of-dimensionality. In this talk, we present a machine-learning framework for the numerical solution of potential MFG and mean field control models.
To enable the solution of high-dimensional instances, we combine formulations from Lagrangian and Eulerian viewpoints, and we leverage recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem but enforce Hamilton Jacobi equations that are derived from an Eulerian formulation. Finally, a neural network parameterization of the MFG/MFC solution helps us to avoid any space discretization.
One significant numerical result is the approximate solution of an optimal transport problem, which is a potential MFG, in as much as 100 dimensions. Our results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing solutions methods, e.g., in density estimation and regularization for medical imaging problems.
This is joint work with Stanley Osher, Wuchen Li, Levon Nurbekyan, and Samy Wu Fung (all from UCLA).
The main reference for this talk is: L. Ruthotto, S. Osher, W. Li, L. Nurbekyan, S. Wu Fung, A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems, arXiv:1912.01825