In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions.

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.

Mean field games (MFGs) have emerged as a powerful framework for modeling interactions in large-scale multi-agent systems. Despite recent advancements in reinforcement learning (RL) for MFGs, existing methods are typically limited to finite spaces or stationary models, hindering their applicability to real-world problems. This paper introduces a novel deep reinforcement learning (DRL) algorithm specifically designed for non-stationary continuous MFGs. The proposed approach builds upon a Fictitious Play (FP) methodology, leveraging DRL for best-response computation and supervised learning for average policy representation. Furthermore, it learns a representation of the time-dependent population distribution using a Conditional Normalizing Flow. To validate the effectiveness of our method, we evaluate it on three different examples of increasing complexity. By addressing critical limitations in scalability and density approximation, this work represents a significant advancement in applying DRL techniques to complex MFG problems, bringing the field closer to real-world multi-agent systems.

Mean-field games (MFGs), introduced independently by Lasry and Lions [39, 40, 41] and by Huang, Caines, and Malham e [32, 31], provide a powerful framework for modeling strategic interactions among a large population of agents, where each agent responds to the aggregate distribution of the population rather than to individual players. Over the past decade, substantial progress has been made in the theoretical development of MFGs, including the well-posedness of equilibria under monotonicity conditions [39], and the rigorous connection to McKean-Vlasov forward-backward stochastic differential equations (FBSDEs) [16] and master equations [14]. A broader exposition of the theory and its historical development can be found in [13, 10, 25, 17]. From a computational perspective, solving MFGs remains challenging due to their intrinsic infinite-dimensional structure arising from the dependence on the evolving population distribution. Classical numerical approaches focus on solving the coupled Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations directly [1]. More recent advances leverage deep learning techniques to approximate the partial differential equation (PDE) systems [49, 9], FBSDEs [19, 24, 28], and even master equations [21, 26]. In parallel, reinforcement learning (RL)-based approaches have attracted growing attention for solving MFGs, motivated by their model-free nature, i.e., the ability to learn optimal strategies directly from observations without requiring explicit knowledge of the system dynamics [27, 48, 5, 4]. We refer interested readers to the recent survey [42]. In this work, we propose the Mean-Field Actor-Critic (MFAC) flow, a learning-based framework for solving MFGs with general distribution dependence.

In such framework, all players are identical, anonymous ( i.e., they are not identifiable) and have symmetric interests.

We introduce Mean-Field Trust Region Policy Optimization (MF-TRPO), a novel algorithm designed to compute approximate Nash equilibria for ergodic Mean-Field Games (MFG) in finite state-action spaces. Building on the well-established performance of TRPO in the reinforcement learning (RL) setting, we extend its methodology to the MFG framework, leveraging its stability and robustness in policy optimization. Under standard assumptions in the MFG literature, we provide a rigorous analysis of MF-TRPO, establishing theoretical guarantees on its convergence. Our results cover both the exact formulation of the algorithm and its sample-based counterpart, where we derive high-probability guarantees and finite sample complexity. This work advances MFG optimization by bridging RL techniques with mean-field decision-making, offering a theoretically grounded approach to solving complex multi-agent problems.

Our friend and mentor Peter Caines has, together with his colleagues, created new foundations for studying collective dynamics in complex systems. Of particular inspiration to us has been his pioneering work in mean-field games (MFGs) launched two decades ago [10, 24, 25], and the related field of mean-field control. Peter pointed the way to both formulate and solve the problem of collective dynamics arising in a large population of heterogeneous dynamical systems. In this paper we survey some elements of MFGs within the context of controlled coupled oscillators. We begin by introducing a model for a single oscillator: dθ(t) = (ω + u(t)) dt + σ dξ(t), mod 2π (1) where θ(t) [0, 2π) is the phase of the oscillator at time t, ω is the nominal frequency with units of radiansper-second, {ξ(t): t 0} is a standard Wiener process, and u(t) is a control signal whose interpretation depends on the context. Unless otherwise noted, the SDEs are interpreted in their Itô form.

Large agent networks are abundant in applications and nature and pose difficult challenges in the field of multi-agent reinforcement learning (MARL) due to their computational and theoretical complexity. While graphon mean field games and their extensions provide efficient learning algorithms for dense and moderately sparse agent networks, the case of realistic sparser graphs remains largely unsolved. Thus, we propose a novel mean field control model inspired by local weak convergence to include sparse graphs such as power law networks with coefficients above two. Besides a theoretical analysis, we design scalable learning algorithms which apply to the challenging class of graph sequences with finite first moment. We compare our model and algorithms for various examples on synthetic and real world networks with mean field algorithms based on Lp graphons and graphexes. As it turns out, our approach outperforms existing methods in many examples and on various networks due to the special design aiming at an important, but so far hard to solve class of MARL problems.