提出可处理不规则域的函数空间扩散模型，实现分辨率无关与复杂几何建模。
GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains
- 用有限元函数表示图卷积核，自然支持非结构化网格
- 在非凸、多连通域上保持分辨率不变性，生成高保真函数分布
- 适合需处理复杂几何形状的科学计算与逆问题场景
基于得分的无限维函数空间扩散模型为函数值数据建模提供了数学上严谨的框架，具有分辨率不变性和处理不规则离散化的优点。然而，现有实现未能充分实现这些优势。如傅里叶神经算子等现有主干网络常偏向规则网格，难以泛化到复杂拓扑域。本文提出一种新型函数空间扩散模型架构，将广义图卷积核表示为有限元函数，使模型能自然处理非结构化网格和复杂几何形状。通过一系列无条件与条件采样实验，在多种几何结构（包括非凸及多连通域）上验证了该方法的有效性。结果表明，所提方法保持了分辨率不变性，并在非平凡几何上高质量捕捉了函数分布。
原文摘要 · Abstract (English)
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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