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This paper provides rigorous mathematical foundations for score-based diffusion models by establishing sharp stability estimates and convergence rates within a partial differential equation framework. By connecting entropy stability methods to the Li–Yau differential inequality, it clarifies how exact-score dynamics concentrate on data support without necessarily recovering the full distribution. These theoretical insights are crucial for understanding the limits of generative fidelity versus diversity in early stopping regimes.
A PDE Perspective on Generative Diffusion Models
arXiv:2511.05940v3 Announce Type: replace-cross Abstract: Score-based diffusion models have emerged as a powerful class of generative methods, with successful applications across diverse domains. Despite their empirical success, their mathematical foundations remain only partially understood, particularly regarding the stability and consistency of the stochastic and partial differential equations underlying their dynamics. In this work, we develop a partial differential equation (PDE) framework for score-based diffusion processes associated with the heat flow of a compactly supported data measure. After establishing weak well-posedness on positive-time intervals, we derive a sharp one-sided divergence bound for the score and use it to obtain sharp uniform $L^p$-stability estimates for the generative Fokker--Planck equation. This bound is the Euclidean form of the Li--Yau differential inequality. Through entropy stability methods, we prove that the generated laws converge in the $2$-Wasserstein distance at rate $\sqrt t$ as $t\to0^+$ to a probability measure supported on the data support. This holds for both deterministic and stochastic generation, for positive terminal densities with finite absolute entropy and second moment. In the deterministic case, almost every trajectory converges to a point of the data support; for empirical data, we further obtain a trajectory-wise square-root rate. These results clarify how exact-score dynamics concentrate on the data support without necessarily recovering the data distribution. An explicitly solvable case identifies the limiting weights and their dependence on Gaussian initialization, while a relative-entropy estimate quantifies the effect of score approximation. Together, these findings connect the stability and concentration mechanisms with score learning and early stopping, providing a mathematical perspective on the balance between imitation fidelity and generative diversity.
Abstract:Score-based diffusion models have emerged as a powerful class of generative methods, with successful applications across diverse domains. Despite their empirical success, their mathematical foundations remain only partially understood, particularly regarding the stability and consistency of the stochastic and partial differential equations underlying their dynamics. In this work, we develop a partial differential equation (PDE) framework for score-based diffusion processes associated with the heat flow of a compactly supported data measure. After establishing weak well-posedness on positive-time intervals, we derive a sharp one-sided divergence bound for the score and use it to obtain sharp uniform $L^p$-stability estimates for the generative Fokker--Planck equation. This bound is the Euclidean form of the Li--Yau differential inequality. Through entropy stability methods, we prove that the generated laws converge in the $2$-Wasserstein distance at rate $\sqrt t$ as $t\to0^+$ to a probability measure supported on the data support. This holds for both deterministic and stochastic generation, for positive terminal densities with finite absolute entropy and second moment. In the deterministic case, almost every trajectory converges to a point of the data support; for empirical data, we further obtain a trajectory-wise square-root rate. These results clarify how exact-score dynamics concentrate on the data support without necessarily recovering the data distribution. An explicitly solvable case identifies the limiting weights and their dependence on Gaussian initialization, while a relative-entropy estimate quantifies the effect of score approximation. Together, these findings connect the stability and concentration mechanisms with score learning and early stopping, providing a mathematical perspective on the balance between imitation fidelity and generative diversity.
Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Analysis of PDEs (math.AP)
arXiv:2511.05940 [math.OC]
(or arXiv:2511.05940v3 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2511.05940 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 356 words
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