Physics-graph-informed neural networks (PGINNs): From local point-wise constraint to global nodal association

Abstract

Physics-Informed Neural Networks (PINNs) and their variants have gained widespread attention as scientific machine learning (SciML) methods for solving Partial Differential Equations (PDEs), but their computational efficiency and accuracy in low-dimensional problems are still lower than that of classical numerical methods, such as Finite Element Method (FEM). This limitation mainly stems from the generalization error of PINNs, whose point-wise loss functional ignores the interactions between neighboring collocation points. To address this challenge, we propose a new Physics-Graph-Informed Neural Networks (abbreviated as PGINNs) framework that combines deep learning with a conservation-consistent nodal network. Compared to traditional PINNs, PGINNs encode two fundamental forms of graph-topological physical information: (1) global node topology operator represented by the sparse incidence matrix, and (2) local edge feature matrix learned through deep neural networks (DNNs). By reconstructing the loss function as a system of algebraic equations learned by the nodal network representation, PGINNs effectively reduce the generalization error while retaining the mesh-free advantage of PINNs. Theoretical analysis shows that the generalization error convergence rate of PGINNs is improved by an order of magnitude compared with that of PINNs. Numerical experiments for complex boundaries and piecewise homogeneous media validate the high accuracy and computational efficiency of the method. This work combines the advantages of both classical numerical analysis and data-driven PDE solvers, establishing a new direction for high-fidelity SciML.