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

TL;DR

This paper introduces a particle-based deep Flow Matching method to efficiently solve high-dimensional mean-field games, overcoming computational challenges and providing convergence guarantees.

Contribution

It proposes a novel particle-based flow matching approach for high-dimensional MFGs with theoretical convergence analysis and practical performance on complex problems.

Findings

Converges to a stationary point sublinearly, with exponential convergence under convexity.

Demonstrates promising results on non-potential MFGs.

Effective in high-dimensional optimal transport problems.

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 MFG computation. In each iteration of our proximal fixed-point 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.