An Operator-Consistent Graph Neural Network for Learning Diffusion Dynamics on Irregular Meshes

TL;DR

This paper introduces an operator-consistent graph neural network that accurately models diffusion dynamics on irregular meshes, maintaining stability and structural fidelity during temporal predictions.

Contribution

It proposes a novel GNN architecture with a consistency loss to enforce PDE structure, improving stability and accuracy on irregular domain diffusion problems.

Findings

Enhanced temporal stability over baseline models

Prediction accuracy approaching traditional PDE solvers

Effective on real-world irregular meshes

Abstract

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.