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

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; thus, the algorithm is simulation-free. Theoretically, we prove that the scheme converges to a fixed point sublinearly in the optimal control setting and linearly with additional convexity assumptions. Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT tasks formulated as MFGs, where the terminal constraint is relaxed.