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.
@article{arxiv.2509.02015,
title = {Second-Order Tensorial Partial Differential Equations on Graphs},
author = {Aref Einizade and Fragkiskos D. Malliaros and Jhony H. Giraldo},
journal= {arXiv preprint arXiv:2509.02015},
year = {2025}
}