  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.
