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

• URL: http://arxiv.org/abs/2512.01172v1
• Date: Mon, 01 Dec 2025 01:04:53 GMT
• Title: High-dimensional Mean-Field Games by Particle-based Flow Matching
• Authors: Jiajia Yu, Junghwan Lee, Yao Xie, Xiuyuan Cheng,
• Abstract summary: Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents.<n>Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles.<n>We propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFGs.
• Score: 18.129646808071893
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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