A PDE Perspective on Generative Diffusion Models

2025-11-08

AI Summary

Generative diffusion models lack a rigorous mathematical foundation, particularly concerning stability and consistency under their stochastic/partial differential equation (PDE) formulations. Method: We establish a PDE-theoretic framework, introducing the Li–Yau differential inequality—novel in diffusion modeling—to derive $L^p$ stability estimates and an entropy stability theory. We analyze the Fokker–Planck equation driven by fractional functions, the reverse stochastic differential equation (SDE), and heat flow evolution. Results: We prove well-posedness of the model dynamics and provide a rigorous guarantee that the reverse process converges to the data manifold at rate $sqrt{t}$. Our analysis quantifies the trade-off between generative capacity and reconstruction fidelity. The framework yields principled guidance for fractional function design, loss formulation, and sampling termination criteria—advancing both theoretical understanding and practical implementation of diffusion models.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

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 $t o 0$. 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.

Problem

• Developing rigorous PDE framework for score-based diffusion processes
• Proving stability and consistency of diffusion model dynamics
• Analyzing trade-off between generative capacity and imitation fidelity

Innovation

• PDE framework for score-based diffusion processes
• Li-Yau inequality for well-posedness and stability
• Entropy methods for data manifold concentration