We demonstrate the versati lity of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections betwe en MFGs and major classes of flow- and diffusion-based generative models by deriving continuous-time normalizing flows and score-based models through different choices of particle dynamics and cost functions. We study the mat hematical structure and properties of each generative model by examining th eir associated MFG optimality conditions, which consist of coupled forward- backward nonlinear partial differential equations (PDEs). We present this f ramework as an MFG laboratory, a platform for experimentation, invention, a nd analysis of generative models. Through this laboratory, we show how MFG structure informs new normalizing flows that robustly learn data distributi ons supported on low-dimensional manifolds. In particular, we show that Was serstein proximal regularizations inform the well-posedness and robustness of generative flows for singular measures, enabling stable training with le ss data and without specialized architectures.

We then apply these principled gen erative models to operator learning, where the goal is to learn solution op erators of differential equations. We present a probabilistic framework tha t reveals certain classes of operator learning approaches, such as in-conte xt operator networks (ICON), as implicitly performing Bayesian inference. I CON computes the mean of the posterior predictive distribution of solution operators conditioned on example condition-solution pairs. By extending ICO N to a generative setting, we enable sampling from the posterior predictive distribution. This provides principled uncertainty quantification for pred icted solutions, demonstrating how mathematical foundations translate to tr ustworthy applications in scientific computing.