GPINN: Physics-informed Neural Network with Graph Embedding

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.