We test the efficiency of applying Geometric Deep Learning to the problems in low-dimensional topology in a certain simple setting. Specifically, we consider the class of 3-manifolds described by plumbing graphs and use Graph Neural Networks (GNN) for the problem of deciding whether a pair of graphs give homeomorphic 3-manifolds. We use supervised learning to train a GNN that provides the answer to such a question with high accuracy. Moreover, we consider reinforcement learning by a GNN to find a sequence of Neumann moves that relates the pair of graphs if the answer is positive. The setting can be understood as a toy model of the problem of deciding whether a pair of Kirby diagrams give diffeomorphic 3- or 4-manifolds.

翻译：我们测试了将几何深度学习应用于低维拓扑中某一简单设定问题的效率。具体而言，我们考虑由管路图描述的三维流形类，并利用图神经网络（GNN）判断一对图是否给出同胚的三维流形。通过监督学习，我们训练了一个能够高精度回答此类问题的GNN。此外，我们采用GNN进行强化学习，在答案为是的情况下，找到连接这对图的诺伊曼移动序列。该设定可理解为判断一对Kirby图是否给出微分同胚的三维或四维流形这一问题的玩具模型。