High-dimensional Mean-Field Games by Particle-based Flow Matching

Jiajia Yu, Junghwan Lee, Yao Xie, Xiuyuan Cheng

Keywords: mean-field games, particle-based, flow matching, simulation-free

TL;DR: A particle-based simulation-free deep flow matching method for solving high-dimensional Mean-Field Games, with proved sublinear convergence.

Abstract:

Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents, which can be formulated as the fixed-point of optimal control problems. They provide a unified framework for a variety of applications, including optimal transport (OT) and generative models. Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles.

In this work, we propose a particle-based deep Flow-Matching (FM) method to tackle high-dimensional deterministic MFG computation.
In each iteration of our proximal best response scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories in a simulation-free manner.
Theoretically, in the optimal control setting, we prove that our scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions.
Our proof uses FM to induce an Eulerian coordinate (density-based) from a Lagrangian one (particle-based), and this also leads to certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficiently regular.
Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT problems cast as MFGs through a relaxed terminal-cost formulation.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 8189