Graph Neural Networks (GNNs) have established themselves as the preferred methodology in a multitude of domains, ranging from computer vision to computational biology, especially in contexts where data inherently conform to graph structures. While many existing methods have endeavored to model GNNs using various techniques, a prevalent challenge they grapple with is the issue of over-smoothing. This paper presents new Graph Neural Network models that incorporate two first-order Partial Differential Equations (PDEs). These models do not increase complexity but effectively mitigate the over-smoothing problem. Our experimental findings highlight the capacity of our new PDE model to achieve comparable results with higher-order PDE models and fix the over-smoothing problem up to 64 layers. These results underscore the adaptability and versatility of GNNs, indicating that unconventional approaches can yield outcomes on par with established techniques.

翻译：图神经网络（GNNs）已在多个领域（从计算机视觉到计算生物学）中确立为首选方法，尤其在数据天然符合图结构的情境中。尽管现有诸多方法尝试利用不同技术对GNNs进行建模，但它们普遍面临过度平滑问题。本文提出了融入两类一阶偏微分方程（PDEs）的新型图神经网络模型。这些模型在不增加复杂度的同时有效缓解了过度平滑问题。实验结果表明，我们提出的新型PDE模型能够达到与高阶PDE模型相当的性能，并在多达64层网络中解决过度平滑问题。这些结果凸显了GNNs的适应性与通用性，表明非传统方法能够取得与成熟技术相媲美的效果。