用图神经网络判断三维流形是否同胚，提速但牺牲精度。
Detecting Homeomorphic 3-manifolds via Graph Neural Networks
- 通过川流图表示流形，利用图神经网络进行同胚判断。
- 在多项式时间内完成判断，比传统方法快但有误差。
- 为数学家提供高效工具，适合拓扑与机器学习交叉研究者。
受6d超共形场论在三流形上紧化后得到的3d N=2超对称量子场论的BPS谱枚举启发，本文研究一类图流形的同胚问题，采用图神经网络（GNN）技术。利用JSJ分解，可从图流形中唯一提取出川流图表示。同胚的图流形可通过一系列冯诺依曼操作关联，算法应用这些操作可在超多项式时间内判断是否同胚。而使用图神经网络则可在多项式时间内完成判断，代价是精度下降。本文构建了一个包含成对川流图及隐藏标签（表示是否同胚）的数据集，通过监督学习训练并评估多种网络架构（如GEN、GCN、GAT、NNConv），比较其在该同胚问题上的表现优劣。
原文摘要 · Abstract (English)
Motivated by the enumeration of the BPS spectra of certain 3d $\mathcal{N}=2$ supersymmetric quantum field theories, obtained from the compactification of 6d superconformal field theories on three-manifolds, we study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. Utilizing the JSJ decomposition, a unique representation via a plumbing graph is extracted from a graph-manifold. Homeomorphic graph-manifolds are related via a sequence of von Neumann moves on this graph; the algorithmic application of these moves can determine if two graphs correspond to homeomorphic graph-manifolds in super-polynomial time. However, by employing Graph Neural Networks (GNNs), the same problem can be addressed, at the cost of accuracy, in polynomial time. We build a dataset composed of pairs of plumbing graphs, together with a hidden label encoding whether the pair is homeomorphic. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers (GEN, GCN, GAT, NNConv), followed by an aggregation layer and a classification layer. We discuss the strengths and weaknesses of the different GNNs for this homeomorphism problem.
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