arXiv:2512.01172v1 Announce Type: new 
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.

تقدم دراسة جديدة طريقة تعتمد على التدفق القائم على الجسيمات لمعالجة التحديات الحسابية للألعاب ذات المجال المتوسط (MFG) عالية الأبعاد، والتي تحلل توازن ناش في أنظمة تحتوي على عدد كبير من الوكلاء المتفاعلين. تقوم هذه الطريقة بتحديث الجسيمات باستخدام معلومات من الدرجة الأولى وتدريب شبكة عصبية للتدفق لمطابقة سرعات المسارات العينة دون محاكاة.

Un nuevo estudio presenta un método de Flow Matching basado en partículas que busca abordar los desafíos computacionales de los Juegos de Campo Medio (MFG) de alta dimensión, que analizan el equilibrio de Nash en sistemas con numerosos agentes interactuantes. Este método actualiza partículas utilizando información de primer orden y entrena una red neuronal de flujo para igualar las velocidades de las trayectorias de muestra sin simulaciones.

Une nouvelle étude présente une méthode de Flow Matching basée sur des particules, visant à relever les défis computationnels des jeux à champ moyen (MFG) de haute dimension, qui analysent l'équilibre de Nash dans des systèmes avec de nombreux agents interagissant. Cette méthode met à jour les particules en utilisant des informations de premier ordre et entraîne un réseau de neurones de flux pour faire correspondre les vitesses des trajectoires d'échantillons sans simulations.

A new study introduces a particle-based deep Flow Matching method aimed at addressing the computational challenges of high-dimensional Mean-Field Games (MFGs), which analyze the Nash equilibrium in systems with numerous interacting agents. This method updates particles using first-order information and trains a flow neural network to match sample trajectory velocities without simulations.

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

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