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

2025-11-30

AI Summary

Solving Nash equilibria in high-dimensional mean-field games (MFGs) remains challenging due to the curse of dimensionality and the difficulty of accurate density estimation. To address this, we propose a simulation-free deep flow matching method that directly aligns Lagrangian particle trajectories with Eulerian velocity fields, bypassing explicit density modeling. This work introduces flow matching to MFGs for the first time, rigorously establishing equivalence between Lagrangian and Eulerian formulations and ensuring iterative convergence. Our framework integrates optimal control principles, proximal fixed-point iteration, and first-order neural network optimization. Experiments demonstrate substantial improvements in both efficiency and accuracy on non-potential MFGs and high-dimensional optimal transport tasks. The approach combines theoretical soundness—guaranteeing well-posedness and convergence—with computational scalability, enabling tractable equilibrium computation in previously infeasible high-dimensional regimes.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Solving high-dimensional mean-field games computationally
• Overcoming computational and analytical obstacles in MFGs
• Addressing non-potential and high-dimensional optimal transport problems

Innovation

• Particle-based flow matching for high-dimensional MFG computation
• Simulation-free neural network training from particle trajectories
• Proximal fixed-point scheme with theoretical convergence guarantees