GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains
This work addresses the challenge of generating function-valued data on irregular domains for applications in scientific computing and engineering, offering a principled approach that maintains resolution invariance.
The paper proposes a graph-based diffusion model architecture that uses finite element functions to represent convolutional kernels, enabling resolution-invariant function generation on irregular domains with complex geometries. The method achieves high fidelity in unconditional and conditional sampling across non-convex and multiply-connected 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 propose a novel architecture for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, enabling the model to naturally handle unstructured meshes and complex geometries. 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.