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).

本研究提出了一种名为去噪扩散算子的数学严谨框架，用于在函数空间中训练扩散模型，将它推广到无限维函数空间的应用，其中前向过程是逐渐扰动输入函数，生成过程是通过积分的函数值 Langevin 动力学实现。