Geometric Neural Diffusion Processes

TL;DR

This paper extends diffusion models to non-Euclidean spaces by incorporating geometric priors and symmetries, enabling effective modeling of complex data like weather fields in infinite dimensions.

Contribution

It introduces a framework for geometric diffusion models with symmetry-equivariant neural networks in infinite-dimensional spaces, applicable to natural science data.

Findings

Successfully models scalar and vector fields in Euclidean and spherical spaces.

Demonstrates scalability with a Langevin-based conditional sampler.

Achieves effective modeling on synthetic and real-world weather data.

Abstract

Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling. Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes. However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces. In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w.r.t. this group. We show that with these conditions, the generative functional model admits the same symmetry. We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.