In this paper, we demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. There is a pervasive sense in the generative modeling community that the various flow and diffusion-based generative models have some foundational common structure and interrelationships. We establish connections between MFGs and major classes of flow and diffusion-based generative models including continuous-time normalizing flows, score-based models, and Wasserstein gradient flows. We derive these three classes of generative models through different choices of particle dynamics and cost functions. Furthermore, we study the mathematical structure and properties of each generative model by studying their associated MFG's optimality condition, which is a set of coupled nonlinear partial differential equations (PDEs). The theory of MFGs, therefore, enables the study of generative models through the theory of nonlinear PDEs. Through this perspective, we investigate the well-posedness and structure of normalizing flows, unravel the mathematical structure of score-based generative modeling, and derive a mean-field game formulation of the Wasserstein gradient flow. From an algorithmic perspective, the optimality conditions of MFGs also allow us to introduce HJB regularizers for enhanced training a broader class of generative models. We present this framework as an MFG laboratory which serves as a platform for revealing new avenues of experimentation and invention of generative models. This laboratory will give rise to a multitude of well-posed generative modeling formulations, providing a consistent theoretical framework upon which numerical and algorithmic tools may be developed.

翻译：在本文中，我们展示了均值场博弈（Mean-Field Games，MFGs）作为解释、增强和设计生成模型的数学框架的通用性。生成建模领域普遍存在这样一种认知：各种基于流和扩散的生成模型具有某种基础性共同结构和内在关联。我们建立了MFGs与三大类基于流和扩散的生成模型（包括连续时间归一化流、基于分数的模型和Wasserstein梯度流）之间的联系。通过选择不同的粒子动力学和代价函数，我们推导出这三类生成模型。此外，我们通过研究各生成模型对应的MFG最优性条件（一组耦合的非线性偏微分方程组）来探讨其数学结构与性质。因此，MFG理论能够借助非线性偏微分方程理论对生成模型进行研究。基于这一视角，我们探讨了归一化流的适定性和结构，揭示了基于分数的生成建模的数学结构，并推导出Wasserstein梯度流的均值场博弈公式。从算法角度看，MFG的最优性条件还使我们能够引入HJB正则化项，以增强更广泛类别生成模型的训练。我们将这一框架称为“MFG实验室”，它作为揭示生成模型实验与发明新途径的平台。该实验室将催生大量适定的生成建模公式，为数值与算法工具的开发提供一致的理论基础。