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.

翻译：平均场博弈（MFGs）研究具有连续相互作用智能体系统的纳什均衡，可表述为最优控制问题的固定点。该框架为多种应用提供了统一的理论基础，包括最优传输（OT）与生成模型。尽管应用广泛，但由于计算与理论分析上的根本性障碍，求解高维平均场博弈仍面临重大挑战。本文提出一种基于粒子的深度流匹配（FM）方法以应对高维平均场博弈的计算问题。在我们的近端固定点迭代方案中，每轮迭代通过一阶信息更新粒子分布，并训练流神经网络以无模拟方式匹配样本轨迹的速度场。在理论层面，针对最优控制设定，我们证明了该方案能以次线性速率收敛至稳定点，并在附加凸性假设下提升至线性（指数）收敛。证明过程中利用流匹配从拉格朗日坐标（基于粒子）导出欧拉坐标（基于密度），这进一步揭示了当欧拉解具有足够正则性时，两种平均场博弈表述间的等价关系。通过松弛终端代价公式将高维最优传输问题转化为平均场博弈，本方法在非势场平均场博弈及高维最优传输问题上均展现出优越性能。