First-order PDES for Graph Neural Networks: Advection And Burgers Equation Models

TL;DR

This work tackles over-smoothing in Graph Neural Networks by introducing first-order PDE-based propagation blocks, notably Advection and Burgers, discretized on graphs via forward Euler updates. It extends the framework with Mixing variants that blend advection with diffusion or wave dynamics through a trainable parameter $\alpha$ and edge-wise weights $D$, achieving robustness up to 64 layers while maintaining comparable performance to higher-order PDE models. Across semi-supervised and fully-supervised node classification and dense shape correspondence tasks, diffusion-dominated mixtures often yield strong results, while wave-focused mixing shines in shape mapping, with the Wave mixing model reaching about $99.9\%$ accuracy on FAUST. The results indicate that simple, physics-inspired first-order dynamics can match established techniques, offering a versatile and scalable alternative for diverse graph-based problems.

Abstract

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.