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

James Rowbottom, Elizabeth Louise Baker, Nick Huang, Ben Adcock, Carola-Bibiane Schönlieb, Alexander Denker

Keywords: Graph networks, inverse problems, infinite-dimensional diffusion models, neural operators

TL;DR: We propose a graph neural operator architecture to learn score-based diffusion models in infinite-dimensional function spaces.

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.

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Submission Number: 70