A PDE Perspective on Generative Diffusion Models

TL;DR

This work provides a rigorous PDE framework for score-based diffusion models by recasting the forward diffusion as the heat equation and the generative process as a backward Fokker–Planck equation driven by the score field $s=\nabla \log u$. A central tool is the Li–Yau inequality which yields sharp lower bounds on $\operatorname{div}(s)$, enabling $L^p$ stability estimates for the backward flow and a contraction property for the KL divergence between forward and backward solutions. Using entropy methods, the authors prove that reverse-time dynamics concentrate on the data manifold for compactly supported data distributions, with a concentration rate of order $\sqrt{t}$ as $t\to 0$ for deterministic generation and finite horizons. The results bridge rigorous PDE analysis with diffusion-model design, offering principled guidelines for score-function construction, loss formulation, and stopping-time selection, and clarifying the trade-off between imitation fidelity and genuine sample generation.

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.

Figures (4)

• Figure 1: Score function (left) and log-density (right) of the heat flow originating from the initial distribution $u_0 = 0.7\,\delta_{-5} + 0.3\,\delta_{0} + 0.1\,\delta_{5}$. The singular behavior predicted by the right-hand side of the Li--Yau estimate as $t \to 0$ is evident from the steep slopes of the score function near the Dirac locations of the initial data. The right panel also shows that the local maxima of the log-densities occur at these Dirac positions as $t\to 0$.
• Figure 2: Score-based generation on the lemniscate dataset: comparison between the true ((A)-(B)) and empirical scores ((C)-(D)), and the effect of the choice of the stopping time $t_{\min} \in \{0.1,\,0.01,\,0.001\}$. The true score ((A)-(B)) corresponds to the solution of the heat equation, given by the convolution with the Gaussian heat kernel, computed by means of numerical quadrature, while the empirical score ((C)-(D)) is obtained from a finite sample of data points via the explicit expression \ref{['eq:score_empirical']}. In all experiments, we set the backward diffusion parameter to be $\epsilon = 0.2$. The generated manifold is obtained as the region encompassing $95\%$ of the $10{,}000$ reverse-time samples, reducing the effect of extreme stochastic trajectories.
• Figure 3: Trajectories of the score-based ODE ($\epsilon=0$) with $u_0 = \tfrac{1}{2}(\delta_{(-1,0)} + \delta_{(1,0)})$. Initial points with $x<0$ converge to $(-1,0)$, those with $x>0$ converge to $(1,0)$, and points with $x=0$ converge to the barycenter $(0,0)$.
• Figure 4: Examples of $\gamma$--Voronoi core of 5 points in $\mathbb{R}^2$. When $\gamma=0$, we recover the classical Voronoi diagram.

Theorems & Definitions (26)

• Theorem 3.1: Energy estimate of the score-based FP equation
• Remark 3.2: Backward well-posedness
• Remark 3.3: Sharpness of the estimate
• Theorem 3.4: Concentration of support
• Remark 3.5: Gradient flows
• Remark 3.6: The inviscid case
• Remark 3.7: Concentration of support
• Remark 3.8: Numerical experiment in Figure \ref{['fig:diffusion_model']}
• Remark 3.9: Empirical score function
• Remark 3.10: Necessity of structural conditions on $v_T$