Second-Order Tensorial Partial Differential Equations on Graphs
This work addresses a domain-specific problem for applications like traffic forecasting by providing a novel theoretical framework for second-order continuous product graph neural networks.
The authors tackled the problem of processing data on multiple interacting graphs by introducing second-order tensorial partial differential equations on graphs (SoTPDEG), which improved information propagation and preserved high-frequency components, leading to superior performance in spatiotemporal traffic forecasting experiments.
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.