(Maths-ia) From Heat Equation to Generative Diffusion Models

Score-based diffusion models have recently emerged as a powerful class of generative models, achieving state-of-the-art performance across a wide range of applications. Despite their empirical success, a rigorous mathematical understanding of their stability and generative mechanisms is largely unexplored. In this work, we introduce a partial differential equation (PDE) framework that provides a solid theoretical foundation for score-based diffusion models. Building upon the Li-Yau differential inequality for the heat flow, we establish well-posedness and sharp energy estimates for the associated score-based Fokker--Planck dynamics, offering a mathematically consistent description of their evolution. Using entropy stability methods, we further prove that the reverse-time dynamics of diffusion models concentrate on the original data manifold for any compactly supported data distribution, a wide class of initial generation distributions, and all finite terminal times. Beyond their theoretical importance, our analysis offers practical insights for diffusion model design, including principled criteria for score-function construction, loss formulation, and stopping-time selection, as well as a quantitative understanding of the trade-off between imitation fidelity and generative capacity. This talk is based on joint work with Enrique Zuazua (FAU Erlangen–Nürnberg).