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：扩散模型最近已成为生成建模的强大框架。它们由正向过程和反向过程组成，其中正向过程用高斯白噪声扰动输入数据，反向过程通过学习评分函数对噪声数据进行去噪以生成样本。尽管取得了巨大成功，但这些模型大多在有限维空间（如欧几里得空间）中构建，这限制了它们在数据具有函数形式领域的应用，例如科学计算和三维几何数据分析。在本研究中，我们提出了一个数学上严谨的框架——去噪扩散算子（Denoising Diffusion Operators, DDOs），用于在函数空间中训练扩散模型。在DDOs中，正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来表述。我们的方法需要对扰动数据分布定义合适的评分函数，这通过将去噪评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固定计算成本生成精确样本，且该成本与数据分辨率无关。我们从理论和数值上验证了该方法在一系列问题上的适用性，包括生成纳维-斯托克斯方程的解（可视为高斯随机场（Gaussian Random Field, GRF）外力作用下的推前分布）。