用图神经网络模拟不规则网格上的扩散过程，更稳定准确。
An Operator-Consistent Graph Neural Network for Learning Diffusion Dynamics on Irregular Meshes
- 结合节点边消息传递与梯度-散度一致性损失，保持离散动态结构耦合。
- 在动态网格和真实扫描表面测试中，预测精度接近克兰克-尼科尔森求解器。
- 适合需物理约束的不规则网格仿真，如多物理场建模与工程计算。
传统数值方法在规则网格上求解偏微分方程（PDE）效率高，但在不规则域上易失稳。实际中，扩散、损伤与愈合等多物理场过程常发生于不规则网格。我们提出一种算子一致的图神经网络（OCGNN-PINN），在物理信息约束下逼近PDE演化过程。该模型通过图关联矩阵强制节点与边间梯度-散度关系的一致性，确保时间推进过程中离散动力学结构耦合。我们在物理驱动的动态网格及真实扫描表面上评估模型，结果表明其相比图卷积与多层感知机基线具有更好的时序稳定性和预测精度，逼近非结构化域上克兰克-尼科尔森求解器的表现。
原文摘要 · Abstract (English)
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.
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