🤖 AI Summary
Existing diffusion models are confined to finite-dimensional Euclidean spaces, rendering them inadequate for modeling functional data—such as PDE solutions and deformation fields—that arise in scientific computing and geometric analysis. This work introduces Denoising Diffusion Operators (DDOs), the first diffusion framework operating directly in infinite-dimensional function spaces. We generalize diffusion processes to the operator level, defining function-valued Langevin dynamics and infinite-dimensional score matching. Our method incorporates a Gaussian process-based forward perturbation scheme, function-space score estimation, and a discretization-aware sampling algorithm. Evaluated on Navier–Stokes solution generation, volcanic InSAR deformation field modeling, and MNIST-SDF synthesis, DDOs achieve high-fidelity functional sampling with computational complexity independent of spatial resolution. This establishes the first systematic generative modeling framework for functions, bridging diffusion modeling with functional analysis and scientific machine learning.
📝 Abstract
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. This work introduces 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 a function-valued annealed 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 independent of the data resolution. We theoretically and numerically verify the applicability of our approach on a set of function-valued problems, including generating solutions to the Navier-Stokes equation viewed as the push-forward distribution of forcings from a Gaussian Random Field (GRF), as well as volcano InSAR and MNIST-SDF.
