  Score-based generative modelling (SGM) has proven to be a very effective
method for modelling densities on finite-dimensional spaces. In this work we
propose to extend this methodology to learn generative models over functional
spaces. To do so, we represent functional data in spectral space to dissociate
the stochastic part of the processes from their space-time part. Using
dimensionality reduction techniques we then sample from their stochastic
component using finite dimensional SGM. We demonstrate our method's
effectiveness for modelling various multimodal datasets.
