Diffusion models have recently emerged as a powerful framework for generative modeling. They consist of a forward process that perturbs input data with Gaussian white noise and a reverse process that learns a score function to generate samples by denoising. Despite their tremendous success, they are mostly formulated on finite-dimensional spaces, e.g. Euclidean, limiting their applications to many domains where the data has a functional form such as in scientific computing and 3D geometric data analysis. In this work, we introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. In DDOs, the forward process perturbs input functions gradually using a Gaussian process. The generative process is formulated by integrating a function-valued Langevin dynamic. Our approach requires an appropriate notion of the score for the perturbed data distribution, which we obtain by generalizing denoising score matching to function spaces that can be infinite-dimensional. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost that is independent of the data resolution. We theoretically and numerically verify the applicability of our approach on a set of problems, including generating solutions to the Navier-Stokes equation viewed as the push-forward distribution of forcings from a Gaussian Random Field (GRF).

翻译：扩散模型近年来已成为生成建模的强大框架。它包含一个前向过程（用高斯白噪声扰动输入数据）和一个逆向过程（学习得分函数并通过去噪生成样本）。尽管取得了巨大成功，但现有模型主要建立在有限维空间（如欧几里得空间）上，限制了其在科学计算和三维几何数据分析等数据具有函数形式的领域中的应用。本文提出了一种名为去噪扩散算子（Denoising Diffusion Operators，DDOs）的数学严谨框架，用于在函数空间中训练扩散模型。在DDOs中，前向过程通过高斯过程逐步扰动输入函数，生成过程则通过积分函数值朗之万动力学实现。该方法需要对扰动数据分布具有适当的得分概念，通过将去噪得分匹配推广到可能无限维的函数空间来实现。我们证明，相应的离散化算法能以与数据分辨率无关的固定成本生成精确样本。通过一系列问题（包括将纳维-斯托克斯方程的解视为高斯随机场外力推前分布而生成的案例），我们从理论和数值上验证了该方法的适用性。