An Operator-Consistent Graph Neural Network for Learning Diffusion Dynamics on Irregular Meshes
This addresses the challenge of modeling diffusion and other multiphysics interactions on irregular domains, which is incremental as it builds on existing graph neural network and physics-informed approaches.
The paper tackled the problem of solving partial differential equations (PDEs) on irregular meshes, where classical methods often become unstable, by developing an operator-consistent graph neural network (OCGNN-PINN) that improved temporal stability and prediction accuracy, approaching the performance of Crank-Nicolson solvers on unstructured domains.
Classical numerical methods solve partial differential equations (PDEs) efficiently on regular meshes, but many of them become unstable on irregular domains. In practice, multiphysics interactions such as diffusion, damage, and healing often take place on irregular meshes. We develop an operator-consistent graph neural network (OCGNN-PINN) that approximates PDE evolution under physics-informed constraints. It couples node-edge message passing with a consistency loss enforcing the gradient-divergence relation through the graph incidence matrix, ensuring that discrete node and edge dynamics remain structurally coupled during temporal rollout. We evaluate the model on diffusion processes over physically driven evolving meshes and real-world scanned surfaces. The results show improved temporal stability and prediction accuracy compared with graph convolutional and multilayer perceptron baselines, approaching the performance of Crank-Nicolson solvers on unstructured domains.