# A mean-field games laboratory for generative modeling

arXiv: https://arxiv.org/abs/2304.13534 · HTML: https://mlsanity.com/paper/2304.13534/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 84 references that are in the map.
- First posted on arXiv 26 Apr 2023.

## Builds on (most cited first, 30 of 44)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)

## Built on by (most cited first, 28 of 28)

- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Transport meets Variational Inference: Controlled Monte Carlo Diffusions](https://mlsanity.com/paper/2307.01050/) (2023)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Score-based generative models are provably robust: an uncertainty quantification perspective](https://mlsanity.com/paper/2405.15754/) (2024)
- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [Neural Sinkhorn Gradient Flow](https://mlsanity.com/paper/2401.14069/) (2024)
- [Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiency](https://mlsanity.com/paper/2505.13499/) (2025)
- [Differential Equations for Continuous-Time Deep Learning](https://mlsanity.com/paper/2401.03965/) (2024)
- [Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems](https://mlsanity.com/paper/2409.16471/) (2024)
- [OT-Transformer: A Continuous-time Transformer Architecture with Optimal Transport Regularization](https://mlsanity.com/paper/2501.18793/) (2025)
- [Stability of Transformers under Layer Normalization](https://mlsanity.com/paper/2510.09904/) (2025)
- [Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributions](https://mlsanity.com/paper/2405.13962/) (2024)
- [Combining Wasserstein-1 and Wasserstein-2 proximals: robust manifold learning via well-posed generative flows](https://mlsanity.com/paper/2407.11901/) (2024)
- [Nonlinear denoising score matching for enhanced learning of structured distributions](https://mlsanity.com/paper/2405.15625/) (2024)
- [Proximal optimal transport divergences](https://mlsanity.com/paper/2505.12097/) (2025)
- [HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models](https://mlsanity.com/paper/2409.09614/) (2024)
- [Generative diffusion models from a PDE perspective](https://mlsanity.com/paper/2501.17054/) (2025)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)
- [Zero-Shot Transferable Solution Method for Parametric Optimal Control Problems](https://mlsanity.com/paper/2509.18404/) (2025)
- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [Learning Paths for Dynamic Measure Transport: A Control Perspective](https://mlsanity.com/paper/2511.03797/) (2025)
- [A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling](https://mlsanity.com/paper/2606.12710/) (2026)
- [Policy Transfer for Continuous-Time Reinforcement Learning: A (Rough) Differential Equation Approach](https://mlsanity.com/paper/2510.15165/) (2025)
- [Learning Generative Dynamics with Soft Law Constraints: A McKean-Vlasov FBSDE Approach](https://mlsanity.com/paper/2605.08928/) (2026)
- [Implicit Neural Optimal Transport via Fixed-Point Optimization](https://mlsanity.com/paper/2605.10792/) (2026)
- [All in One: Generative Modeling as Mean-Field Game Design](https://mlsanity.com/paper/2607.23026/) (2026)
- [Deep-MKV-TS: Path-Dependent McKean--Vlasov Control for Financial Time Series Generation](https://mlsanity.com/paper/2608.19394/) (2026)

## Abstract (arXiv)

We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. In generative flows, a Lagrangian formulation is used where each particle (generated sample) aims to minimize a loss function over its simulated path. The loss, however, is dependent on the paths of other particles, which leads to a competition among the population of particles. The asymptotic behavior of this competition yields a mean-field game. We establish connections between MFGs and major classes of generative flows and diffusions including continuous-time normalizing flows, score-based generative models (SGM), and Wasserstein gradient flows. Furthermore, we study the mathematical properties of each generative model by studying their associated MFG's optimality condition, which is a set of coupled forward-backward nonlinear partial differential equations. The mathematical structure described by the MFG optimality conditions identifies the inductive biases of generative flows. We investigate the well-posedness and structure of normalizing flows, unravel the mathematical structure of SGMs, and derive a MFG formulation of Wasserstein gradient flows. From an algorithmic perspective, the optimality conditions yields Hamilton-Jacobi-Bellman (HJB) regularizers for enhanced training of generative models. In particular, we propose and demonstrate an HJB-regularized SGM with improved performance over standard SGMs. We present this framework as an MFG laboratory which serves as a platform for revealing new avenues of experimentation and invention of generative models.