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 common foundational 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 forward-backward 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 of a broad class of generative models. In particular, we propose and demonstrate an Hamilton-Jacobi-Bellman 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. This laboratory will give rise to a multitude of well-posed generative modeling formulations and will provide a consistent theoretical framework upon which numerical and algorithmic tools may be developed.

翻译：本文展示了平均场博弈（MFGs）作为数学框架在解释、增强和设计生成模型方面的通用性。在生成模型研究领域，人们普遍认为各种基于流和扩散的生成模型具有某种共同的基础结构和相互关联。我们建立了MFGs与主流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成本函数，我们推导出了这三类生成模型。此外，通过研究每个生成模型对应的MFG最优性条件（一组耦合的前向-后向非线性偏微分方程（PDEs）），我们探讨了各类生成模型的数学结构和性质。因此，MFG理论使我们能够通过非线性PDE理论来研究生成模型。基于这一视角，我们考察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，并推导了Wasserstein梯度流的平均场博弈形式。从算法角度看，MFG的最优性条件还使我们能够引入HJB正则化器，以增强广泛生成模型的训练。具体而言，我们提出并验证了一种汉密尔顿-雅可比-贝尔曼（Hamilton-Jacobi-Bellman）正则化SGM，其性能优于标准SGM。我们将该框架呈现为一个MFG实验室，它可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，并为数值与算法工具的开发提供一致的理论框架。