Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose a path integral based graph neural networks (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian to a new transition matrix we call maximal entropy transition (MET) matrix derived from a path integral formalism. Importantly, the diagonal entries of the MET matrix are directly related to the subgraph centrality, thus providing a natural and adaptive pooling mechanism. PAN provides a versatile framework that can be tailored for different graph data with varying sizes and structures. We can view most existing GNN architectures as special cases of PAN. Experimental results show that PAN achieves state-of-the-art performance on various graph classification/regression tasks, including a new benchmark dataset from statistical mechanics we propose to boost applications of GNN in physical sciences.

中文翻译：

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图神经网络的基于路径积分的卷积和池化

图神经网络 (GNN) 将传统神经网络的功能扩展到图结构数据。与 CNN 类似，图卷积和池化的优化设计是成功的关键。借鉴物理学的思想，我们提出了一种基于路径积分的图神经网络 (PAN)，用于图的分类和回归任务。具体来说，我们考虑一个卷积操作，它涉及连接消息发送者和接收者的每条路径，其权重取决于路径长度，对应于最大熵随机游走。它将图拉普拉斯算子推广到一个新的转换矩阵，我们称之为最大熵转换 (MET) 矩阵，该矩阵源自路径积分形式。重要的是，MET 矩阵的对角线项与子图中心性直接相关，从而提供自然和自适应的池化机制。PAN 提供了一个通用框架，可以针对具有不同大小和结构的不同图形数据进行定制。我们可以将大多数现有的 GNN 架构视为 PAN 的特例。实验结果表明，PAN 在各种图分类/回归任务上取得了最先进的性能，包括我们提出的一个来自统计力学的新基准数据集，以促进 GNN 在物理科学中的应用。