提出几何感知的图神经网络表达力评估方法，能区分不同形状的网格。
Geometry-Aware Simplicial Message Passing
- 将顶点坐标融入颜色精炼过程，增强对几何结构的感知能力。
- 理论证明几何感知模型的表达力上限由GSWL测试决定。
- 结合欧拉特征变换，实现对几何复杂体的完整刻画与近似建模。
Weisfeiler--Lehman (WL) 测试及其单纯形扩展（SWL）刻画了消息传递网络的组合表达力，但对几何信息不敏感——具有相同连通性但不同嵌入的网格无法区分。本文提出几何单纯形 Weisfeiler--Lehman（GSWL）测试，将顶点坐标纳入颜色精炼过程，以感知几何结构。我们证明：(i) 几何感知单纯形消息传递方案的表达力被 GSWL 严格上界；(ii) 存在参数使得这些方案在任意有限几何单纯形复形族上可达到 GSWL 的判别能力。结合欧拉特征变换（ECT），一种几何单纯形复形的完备不变量，本工作构建了完整的几何表达力表征体系，并提供近似框架。在合成数据和网格数据集上的实验验证了理论结果，清晰展现了从组合模型到几何感知模型的表达力层次结构。
原文摘要 · Abstract (English)
The Weisfeiler--Lehman (WL) test and its simplicial extension (SWL) characterize the combinatorial expressivity of message passing networks, but they are blind to geometry, i.e., meshes with identical connectivity but different embeddings are indistinguishable. We introduce the Geometric Simplicial Weisfeiler--Lehman (GSWL) test, which incorporates vertex coordinates into color refinement for geometric simplicial complexes. In addition, we show that (i) the expressivity of geometry-aware simplicial message passing schemes is bounded above by GSWL, and (ii) that there exist parameters such that the discriminating power of GSWL is matched by these schemes on any fixed finite family of geometric simplicial complexes. Combined with the Euler Characteristic Transform (ECT), a complete invariant for geometric simplicial complexes, this yields a geometric expressivity characterization together with an approximation framework. Experiments on synthetic and mesh datasets serve to validate our theory, showing a clear hierarchy from combinatorial to geometry-aware models.
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