Adaptivity and convergence of probability flow ODEs in diffusion generative models

Abstract Score-based generative models, which transform noise into data by learning to reverse a diffusion process, have become a cornerstone of modern generative artificial intelligence. This paper contributes to establishing theoretical guarantees for the probability flow ordinary differential equation (ODE), a widely used diffusion-based sampler known for its practical efficiency. While a number of prior works address its general convergence theory, it remains unclear whether the probability flow ODE sampler can adapt to the low-dimensional structures commonly present in natural image data. We demonstrate that, with accurate score function estimation, the probability flow ODE sampler achieves a convergence rate of $O(k/T)$ in total variation distance (ignoring logarithmic factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of iterations. This dimension-free convergence rate improves upon existing results that scale with the typically much larger ambient dimension, highlighting the ability of the probability flow ODE sampler to exploit intrinsic low-dimensional structures in the target distribution for faster sampling.