Data-driven simulation of physical systems has recently kindled significant attention, where many neural models have been developed. In particular, mesh-based graph neural networks (GNNs) have demonstrated significant potential in predicting spatiotemporal dynamics across arbitrary geometric domains. However, the existing node-edge message passing mechanism in GNNs limits the model's representation learning ability. In this paper, we proposed a cell-embedded GNN model (aka CeGNN) to learn spatiotemporal dynamics with lifted performance. Specifically, we introduce a learnable cell attribution to the node-edge message passing process, which better captures the spatial dependency of regional features. Such a strategy essentially upgrades the local aggregation scheme from the first order (e.g., from edge to node) to a higher order (e.g., from volume to edge and then to node), which takes advantage of volumetric information in message passing. Meanwhile, a novel feature-enhanced block is designed to further improve the performance of CeGNN and relieve the over-smoothness problem, via treating the latent features as basis functions. The extensive experiments on various PDE systems and one real-world dataset demonstrate that CeGNN achieves superior performance compared with other baseline models, particularly reducing the prediction error with up to 1 orders of magnitude on several PDE systems.

翻译：数据驱动的物理系统仿真近年来引起了广泛关注，众多神经模型相继被提出。其中，基于网格的图神经网络在预测任意几何域上的时空动力学方面展现出巨大潜力。然而，现有图神经网络中节点-边消息传递机制限制了模型的表示学习能力。本文提出一种单元嵌入图神经网络模型（简称CeGNN），以提升的性能学习时空动力学。具体而言，我们在节点-边消息传递过程中引入可学习的单元属性，从而更好地捕捉区域特征的空间依赖性。该策略本质上将局部聚合方案从一阶（如从边到节点）升级到更高阶（如从体积到边再到节点），充分利用了消息传递中的体积信息。同时，我们设计了一种新颖的特征增强模块，通过将潜在特征视为基函数，进一步提升CeGNN的性能并缓解过度平滑问题。在多种偏微分方程系统和一个真实数据集上的大量实验表明，CeGNN相比其他基线模型具有更优越的性能，尤其在多个偏微分方程系统上将预测误差降低了最多1个数量级。