# High-dimensional Mean-Field Games by Particle-based Flow Matching arXiv: https://arxiv.org/abs/2512.01172 ยท HTML: https://mlsanity.com/paper/2512.01172/ ## Key facts - Cited by 3 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 31 of its 47 references that are in the map. - First posted on arXiv 1 Dec 2025. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 30 of 31) - [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015) - [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Generative Visual Manipulation on the Natural Image Manifold](https://mlsanity.com/paper/1609.03552/) (2016) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [Learning to Discover Cross-Domain Relations with Generative Adversarial Networks](https://mlsanity.com/paper/1703.05192/) (2017) - [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024) - [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020) - [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022) - [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019) - [Fictitious Play for Mean Field Games: Continuous Time Analysis and Applications](https://mlsanity.com/paper/2007.03458/) (2020) - [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023) - [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022) - [Learning in Mean Field Games: the Fictitious Play](https://mlsanity.com/paper/1507.06280/) (2015) - [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020) - [On the Approximation of Cooperative Heterogeneous Multi-Agent Reinforcement Learning (MARL) using Mean Field Control (MFC)](https://mlsanity.com/paper/2109.04024/) (2021) - [A General Framework for Learning Mean-Field Games](https://mlsanity.com/paper/2003.06069/) (2020) - [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022) - [Mean-Field-Type Games in Engineering](https://mlsanity.com/paper/1605.03281/) (2016) - [Computing high-dimensional optimal transport by flow neural networks](https://mlsanity.com/paper/2305.11857/) (2023) - [A deep learning algorithm for computing mean field control problems via forward-backward score dynamics](https://mlsanity.com/paper/2401.09547/) (2024) - [An Efficient On-Policy Deep Learning Framework for Stochastic Optimal Control](https://mlsanity.com/paper/2410.05163/) (2024) - [Convergence of Actor-Critic Learning for Mean Field Games and Mean Field Control in Continuous Spaces](https://mlsanity.com/paper/2511.06812/) (2025) - [Machine Learning architectures for price formation models](https://mlsanity.com/paper/2204.03968/) (2022) - [Learning Mean-Field Games through Mean-Field Actor-Critic Flow](https://mlsanity.com/paper/2510.12180/) (2025) - [Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence](https://mlsanity.com/paper/2011.07117/) (2020) - [Convergence Analysis and Acceleration of Fictitious Play for General Mean-Field Games via the Best Response](https://mlsanity.com/paper/2411.07989/) (2024) ## Built on by (most cited first, 3 of 3) - [Assured autonomy: How operations research powers and orchestrates generative AI systems](https://mlsanity.com/paper/2512.23978/) (2025) - [Generative models for decision-making under distributional shift](https://mlsanity.com/paper/2604.04342/) (2026) - [Self-supervised In-context Operator Learning for Stochastic Mean-Field Control](https://mlsanity.com/paper/2608.18282/) (2026) ## Abstract (arXiv) 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.