Second-Order Tensorial Partial Differential Equations on Graphs

Abstract: Processing data that lies on multiple interacting (product) graphs is increasingly important in practical applications, yet existing methods are mostly restricted to discrete graph filtering. Tensorial partial differential equations on graphs (TPDEGs) offer a principled framework for modeling such multidomain data in a continuous setting. However, current continuous approaches are limited to first-order derivatives, which tend to dampen high-frequency signals and slow down information propagation. This makes these TPDEGs-based approaches less effective for capturing complex, multi-scale, and heterophilic structures. In this paper, we introduce second-order TPDEGs (So-TPDEGs) and propose the first theoretically grounded framework for second-order continuous product graph neural networks. Our approach leverages the separability of cosine kernels in Cartesian product graphs to implement efficient spectral decomposition, while naturally preserving high-frequency information. We provide rigorous theoretical analyses of stability under graph perturbations and over-smoothing behavior regarding spectral properties. Our theoretical results establish a robust foundation for advancing continuous graph learning across multiple practical domains.