Score-Based Diffusion Models in Function Space

Modern machine learning applications increasingly rely on data represented as continuous functions rather than discrete vectors, including fluid dynamics simulations, climate modeling, medical imaging, and 3D geometric surfaces. Generative diffusion models have demonstrated strong performance across standard image and audio tasks, but their standard formulations operate on finite-dimensional Euclidean spaces. When applied to functional data, standard diffusion models either require retraining for every change in grid resolution or suffer severe performance degradation at finer spatial scales. These limitations restrict the practical deployment of generative artificial intelligence in scientific computing and physical engineering domains.

The main objective of the article is to develop and evaluate Denoising Diffusion Operators (DDOs), a mathematically rigorous framework that extends score-based diffusion modeling to infinite-dimensional function spaces. The article demonstrates that this formulation allows neural operators to generate accurate functional data samples with computational cost and parameter capacity that remain invariant to the underlying spatial discretization.

The authors approach the problem by formulating the forward noise perturbation and reverse generative sampling processes directly on Hilbert spaces using trace-class Gaussian processes rather than standard Gaussian white noise. They extend finite-dimensional score matching theory by defining the score operator as the Fréchet derivative of the logarithmic density with respect to a Gaussian reference measure. The resulting score operator is approximated using resolution-invariant neural operator architectures, such as Fourier Neural Operators and U-shaped Neural Operators. The framework is evaluated across synthetic Gaussian mixture models, fluid flow simulations governed by the non-linear Navier-Stokes equations, satellite radar interferometry of volcanic terrain, signed distance functions of handwritten digits (MNIST-SDF), and a Darcy flow Bayesian inverse problem.

The analysis reveals several primary findings. First, employing trace-class, structured Gaussian noise ensures strict discretization invariance; while white-noise diffusion models suffer rapidly growing error when moving from coarse grids to resolutions of 256 or higher, trace-class models maintain uniform error across all resolutions. Second, neural operators trained on low-resolution data generalize seamlessly to higher resolutions without retraining. For example, a model trained on 128x128 Navier-Stokes fluid simulations generated high-fidelity physical solutions at 1024x1024 resolution, accurately matching critical physical statistics such as energy spectra and kinetic energy density. Third, in 2D image-based shape benchmarks, the proposed framework outperformed alternative functional generative baselines, improving image quality metrics and maintaining a higher sample recall across multiple resolutions (achieving a Fréchet Inception Distance of 2.74 at 64x64 compared to 3.41 for adversarial operators and 35.09 for continuous-time baselines). Finally, the framework successfully solves Bayesian inverse problems, producing posterior distribution statistics that closely match computationally intensive benchmark simulations.

These findings imply that generative modeling can be deployed directly on physical and scientific systems without tying model architecture to a single computational mesh. Organizations can train diffusion models once on lower-resolution, lower-cost datasets and reliably deploy them for high-resolution simulation, super-resolution tasks, and uncertainty quantification in scientific workflows. This reduces training compute costs and mitigates the mode collapse and training instability common in adversarial operator methods.

For practical implementation, engineering and data science teams working on scientific generative modeling should transition from standard white-noise perturbations to trace-class Gaussian processes with appropriate spatial smoothness parameters. When data distributions lack sufficient smoothness relative to the noise process, teams should apply smoothing or blurring operators to preserve mathematical consistency. When designing network architectures, practitioners should balance spectral convolutions to preserve resolution invariance while monitoring potential high-frequency ringing artifacts.

A primary limitation of this work is that spectral convolution layers can introduce subtle boundary ringing artifacts when cutting off high-frequency components on non-periodic data. Additionally, achieving measure equivalence requires the noise process covariance to be appropriately matched to the smoothness of the underlying data distribution. Within the stated theoretical conditions, the evidence strongly supports DDO as a stable, resolution-independent foundation for generative modeling on function spaces.