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：我们测试了几何深度学习在低维拓扑问题中的效率，基于一个特定简单设定。具体而言，我们考虑由管道图描述的三维流形类，并利用图神经网络（GNN）解决一对图是否给出同胚三维流形的判定问题。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，我们采用基于GNN的强化学习，在答案为肯定时寻找连接这对图的纽曼移动序列。该设定可视为判定一对Kirby图是否给出微分同胚三维或四维流形问题的玩具模型。