Zhengjiang Lin

Daniel Zhengyu Huang, Jiaoyang Huang, Zhengjiang Lin

Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to $L^2$-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by $\mathcal{O}(d^{3/4}δ^{1/2})$ in the continuous time level, where $d$ denotes the data dimension and $δ$ represents the $L^2$-score matching error. For practical implementations using a $p$-th order Runge-Kutta integrator with step size $h$, we establish error bounds of $\mathcal{O}(d^{3/4}δ^{1/2} + d\cdot(dh)^p)$ at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.

Daniel Zhengyu Huang, Jiaoyang Huang, Zhengjiang Lin

Diffusion probabilistic models generate samples by learning to reverse a noise-injection process that transforms data into noise. A key development is the reformulation of the reverse sampling process as a deterministic probability flow ordinary differential equation (ODE), which allows for efficient sampling using high-order numerical solvers. Unlike traditional time integrator analysis, the accuracy of this sampling procedure depends not only on numerical integration errors but also on the approximation quality and regularity of the learned score function, as well as their interaction. In this work, we present a rigorous convergence analysis of deterministic samplers derived from probability flow ODEs for general forward processes with arbitrary variance schedules. Specifically, we develop and analyze $p$-th order (exponential) Runge-Kutta schemes, under the practical assumption that the first and second derivatives of the learned score function are bounded. We prove that the total variation distance between the generated and target distributions can be bounded as \begin{align*} O\bigl(d^{\frac{7}{4}}\varepsilon_{\text{score}}^{\frac{1}{2}} +d(dH_{\max})^p\bigr), \end{align*} where $\varepsilon^2_{\text{score}}$ denotes the $L^2$ error in the score function approximation, $d$ is the data dimension, and $H_{\max}$ represents the maximum solver step size. Numerical experiments on benchmark datasets further confirm that the derivatives of the learned score function are bounded in practice.