  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.
