Second-Order Tensorial Partial Differential Equations on Graphs

ABSTRACT Processing data on multiple interacting graphs is crucial for many applications, but existing approaches rely mostly on discrete filtering or first-order continuous models, dampening high frequencies and slow information propagation. In this paper, we introduce second-order tensorial partial differential equations on graphs (SoTPDEG) and propose the first theoretically grounded framework for second-order continuous product graph neural networks (GNNs). Our method exploits the separability of cosine kernels in Cartesian product graphs to enable efficient spectral decomposition while preserving high-frequency components. We further provide rigorous over-smoothing and stability analysis under graph perturbations, establishing a solid theoretical foundation. Experimental results on spatiotemporal traffic forecasting illustrate the superiority over the compared methods. Index T erms-- Tensorial partial differential equations, graph neural networks, over-smoothing, traffic forecasting, stability 1. INTRODUCTION Tensors [1], which generalize matrices to higher dimensions, arise in diverse domains such as brain analysis [2, 3], rec-ommender systems [4], and spatiotemporal forecasting [5, 6].