Processing multidomain data defined on multiple graphs holds significant potential in various practical applications in computer science. However, current methods are mostly limited to discrete graph filtering operations. Tensorial partial differential equations on graphs (TPDEGs) provide a principled framework for modeling structured data across multiple interacting graphs, addressing the limitations of the existing discrete methodologies. In this paper, we introduce Continuous Product Graph Neural Networks (CITRUS) that emerge as a natural solution to the TPDEG. CITRUS leverages the separability of continuous heat kernels from Cartesian graph products to efficiently implement graph spectral decomposition. We conduct thorough theoretical analyses of the stability and over-smoothing properties of CITRUS in response to domain-specific graph perturbations and graph spectra effects on the performance. We evaluate CITRUS on well-known traffic and weather spatiotemporal forecasting datasets, demonstrating superior performance over existing approaches. The implementation codes are available at https://github.com/ArefEinizade2/CITRUS.

翻译：处理定义在多个图上的多领域数据在计算机科学的诸多实际应用中具有重要潜力。然而，当前方法大多局限于离散的图滤波操作。图上的张量偏微分方程为跨多个交互图的结构化数据建模提供了一个原则性框架，解决了现有离散方法的局限性。本文提出连续乘积图神经网络（CITRUS），它作为TPDEG的一个自然解而出现。CITRUS利用笛卡尔图积中连续热核的可分离性，高效实现图谱分解。我们对CITRUS在特定领域图扰动下的稳定性及其过平滑特性，以及图谱效应对性能的影响进行了深入的理论分析。我们在知名的交通和天气时空预测数据集上评估CITRUS，结果表明其性能优于现有方法。实现代码可在 https://github.com/ArefEinizade2/CITRUS 获取。