arXiv cs.AI· Kang Liu, Enrique Zuazua·· 9 天前AI 评分22
生成扩散模型的偏微分方程视角
A PDE Perspective on Generative Diffusion Models
AI 导读
研究者为基于 score 的扩散模型建立了偏微分方程框架，针对紧支撑数据测度的热流，在正时间区间上证明了弱适定性，并推导出 score 的尖锐单侧散度界，进而得到生成 Fokker–Planck 方程的尖锐一致 L^p 稳定性估计。通过熵稳定性方法，他们证明生成分布以 √t 速率在 2-Wasserstein 距离下收敛到支撑于数据支撑集上的概率测度，该结论对确定性和随机生成均成立。
正文 · 原文
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.
| Comments: | 39 pages, 4 figures |
| Subjects: | Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Analysis of PDEs (math.AP) |
| MSC classes: | 34D05, 35B35, 35Q68, 35Q84, 68T99 |
| Cite as: | arXiv:2511.05940 [math.OC] |
| (or arXiv:2511.05940v3 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2511.05940 arXiv-issued DOI via DataCite |
Submission history
From: Kang Liu [view email]
[v1]
Sat, 8 Nov 2025 09:19:25 UTC (1,905 KB)
[v2]
Tue, 12 May 2026 15:40:21 UTC (1,905 KB)
[v3]
Wed, 7 Oct 2026 20:12:33 UTC (1,475 KB)
来源：arXiv cs.AI · arxiv.org