Title: High-dimensional Mean-Field Games by Particle-based Flow Matching
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: mean-field games, particle-based, flow matching, simulation-free

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 problems, including optimal transport (OT) and normalizing flows.
Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to the optimal control fixed-point structure.
In this work, we propose a particle-based deep flow matching method for efficiently solving high-dimensional MFGs. 
At each iteration, particles are updated using first-order information,
and a flow network is trained by matching the velocity field on samples and thus the algorithm is simulation-free.
Theoretically, we prove that the scheme converges to a fixed point sublinearly in the potential MFG setting and linearly in the optimal control setting.
Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT tasks formulated as MFGs, where the terminal constraint is relaxed.

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