GRIFDIR: Graph Resolution-Invariant Diffusion Models over Irregular Domains

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We introduce an architecture, GRIFDIR, for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, allowing the score network to operate directly on unstructured meshes over domains of arbitrary shape. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.