A PDE Perspective on Generative Diffusion Models

TL;DR

This paper develops a rigorous PDE framework for score-based diffusion models, providing theoretical insights into their stability, convergence, and data manifold concentration, which informs better model design.

Contribution

It introduces a PDE-based analysis of diffusion models, establishing well-posedness, stability, and data manifold concentration results, bridging theory and practical model design.

Findings

Proves well-posedness and stability of the PDEs governing diffusion models.

Shows diffusion trajectories concentrate on the data manifold with rate √t.

Provides criteria for score-function construction and stopping-time selection.

Abstract

Score-based diffusion models have emerged as a powerful class of generative methods, achieving state-of-the-art performance across diverse domains. Despite their empirical success, the mathematical foundations of those models remain only partially understood, particularly regarding the stability and consistency of the underlying stochastic and partial differential equations governing their dynamics. In this work, we develop a rigorous partial differential equation (PDE) framework for score-based diffusion processes. Building on the Li--Yau differential inequality for the heat flow, we prove well-posedness and derive sharp $L^p$-stability estimates for the associated score-based Fokker--Planck dynamics, providing a mathematically consistent description of their temporal evolution. Through entropy stability methods, we further show that the reverse-time dynamics of diffusion models concentrate on the data manifold for compactly supported data distributions and a broad class of initialization schemes, with a concentration rate of order $\sqrt{t}$ as $t \to 0$. These results yield a theoretical guarantee that, under exact score guidance, diffusion trajectories return to the data manifold while preserving imitation fidelity. Our findings also provide practical insights for designing diffusion models, including principled criteria for score-function construction, loss formulation, and stopping-time selection. Altogether, this framework provides a quantitative understanding of the trade-off between generative capacity and imitation fidelity, bridging rigorous analysis and model design within a unified mathematical perspective.