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

TL;DR

This paper introduces a graph-based diffusion model architecture that leverages finite element functions to handle irregular domains and complex geometries, maintaining resolution invariance in function space modeling.

Contribution

It presents a novel finite element-based graph convolutional kernel approach for diffusion models, improving generalization to unstructured meshes and complex geometries.

Findings

Model maintains resolution invariance across diverse geometries.

Achieves high fidelity in sampling on non-convex and multiply-connected domains.

Outperforms traditional methods biased towards regular grids.

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.