This work proposes a Physics-informed Neural Network framework with Graph Embedding (GPINN) to perform PINN in graph, i.e. topological space instead of traditional Euclidean space, for improved problem-solving efficiency. The method integrates topological data into the neural network's computations, which significantly boosts the performance of the Physics-Informed Neural Network (PINN). The graph embedding technique infuses extra dimensions into the input space to encapsulate the spatial characteristics of a graph while preserving the properties of the original space. The selection of these extra dimensions is guided by the Fiedler vector, offering an optimised pathologic notation of the graph. Two case studies are conducted, which demonstrate significant improvement in the performance of GPINN in comparison to traditional PINN, particularly in its superior ability to capture physical features of the solution.

本文提出了一种基于图嵌入的物理信息神经网络框架（GPINN），使用拓扑空间而非传统的欧几里得空间来执行PINN，以提高问题解决效率。该框架将拓扑数据集成到神经网络的计算中，显著提高了PINN的性能。通过Fiedler向量引导选择额外的维度，将额外的维度注入输入空间，以封装图的空间特征，同时保留原空间的属性。此外，本文进行了两个案例研究，证明GPINN在与传统PINN相比，特别是在捕获解的物理特征方面具有显著的性能优势。

GPINN: 带有图嵌入的物理知识神经网络