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

TL;DR

This paper tackles the computational challenge of high-dimensional mean-field games by proposing a particle-based proximal fixed-point method that alternates between updating particle trajectories and a neural velocity field via Flow Matching. By reformulating MFGs in Lagrangian coordinates and using FM to reconcile particle trajectories with a consistent Eulerian flow, the approach yields a trajectory-disentangling mechanism that links particle dynamics to density evolution and enables simulation-free optimization. The authors establish sublinear convergence in the general setting and linear convergence under convexity, and demonstrate the method on a 2D OT interpolation task, a non-potential MFG, and image-to-image translation framed as a relaxed OT problem, highlighting scalability to high dimensions and practical applications in generative modeling and transport. The work also discusses equivalence between Eulerian and Lagrangian formulations under regularity and outlines future directions, including memory efficiency, broader theoretical guarantees, and broader applications.

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.

Figures (4)

• Figure 1: Illustration of the trajectory disentangle effect by Flow Matching (FM). (a) and (b) show the sample trajectories at the beginning of training and after 20 epochs of training, respectively, where "optimized particles" stands for trajectories after a particle update, and "velocity field" shows trajectories resampled from a learned velocity field by FM, see Algorithm \ref{['alg: gradient descent']}. Theoretically, we prove that FM disentangles the trajectories and reduces the dynamic cost, while leaving the interaction and terminal costs unchanged.
• Figure 2: Intermediate distributions between $4 \times 4$ checkerboard and an isotropic Gaussian solved by the proposed algorithm.
• Figure 3: Numerical results for the non-potential MFG in Section \ref{['subsec:num non-pot']}. (a) shows convergence of the algorithm. (b) and (c) illustrate that the algorithm output aligns with the physical intuition of a Nash equilibrium.
• Figure 4: Randomly selected three OT trajectories for image-to-image translation of (a) handbags $\to$ shoes and (b) CelebA male $\to$ CelebA female.

Theorems & Definitions (17)

• Proposition 2.1
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• Lemma 4.4
• Lemma 4.5
• Theorem 4.6
• Definition B.1: Absolutely continuous curves
• Definition B.2: First variation cardaliaguet2017ficplay
• proof