GPINN: Physics-Informed Neural Network with Graph Embedding

Solving partial differential equations (PDEs) is of great importance in numerous fields including physics, engineering, finance, and scientific computing. Physics-Informed Neural Networks (PINNs) have gained interest due to their ability to solve PDEs in the strong form, which differs from the traditional methods like the Finite Element Method (FEM) which exhibits the weak form. However, the PINN methods normally operate in the Euclidean space only and lack spatial contextual awareness. This deficiency leads to distance inequality, whereby the Euclidean distance might not align with the actual physical distance. Such misalignment may lead the model to produce high errors with no physical meanings. In this study, we propose the Physics-Informed Neural Network with Graph Embedding (GPINN), which utilises the eigenvectors of graph Laplacian to transform the input space from a pure Euclidean space to a joint Euclidean and topological (graph-based) space and make the model spatial-context aware. Two case studies are conducted to model heat propagation and linear elasticity and show the enhancement from PINN to GPINN.