GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains

2026-05-05

AI Summary

Existing function-space diffusion models struggle to achieve resolution invariance and strong generalization on unstructured grids and irregular geometries. This work proposes a novel function-space diffusion architecture that, for the first time, integrates finite element methods with graph convolution. By expressing generalized graph convolutional kernels in terms of finite element basis functions, the approach naturally supports function generation over domains with arbitrary topology. The method enables high-fidelity unconditional and conditional sampling on complex geometric regions—including non-convex and multiply connected domains—and substantially enhances cross-resolution generalization performance.

Abstract

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.

Problem

• function-space diffusion models
• resolution invariance
• irregular domains
• complex geometries
• unstructured meshes

Innovation

• function-space diffusion models
• resolution invariance
• finite element methods
• graph convolutional kernels
• irregular domains