Second-Order Tensorial Partial Differential Equations on Graphs

2025/09/02 by Einizade, Aref, Malliaros, Fragkiskos D., Giraldo, Jhony H.
#FOS: Computer and information sciences #Machine Learning (cs.LG)

paper · doi:10.48550/arxiv.2509.02015

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.

Citations

• Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations
• Graph Wave Networks
• Continuous Product Graph Neural Networks
• Graph Deep Learning for Time Series Forecasting
• From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond
• Graph Tensor Networks: An Intuitive Framework for Designing Large-Scale Neural Learning Systems on Multiple Domains
• A Survey on Oversmoothing in Graph Neural Networks
• ProductGraphSleepNet: Sleep Staging using Product Spatio-Temporal Graph Learning with Attentive Temporal Aggregation
• TIDE: Time Derivative Diffusion for Deep Learning on Graphs
• Learning Product Graphs from Spectral Templates
• Scalable Spatiotemporal Graph Neural Networks
• Graph-Time Convolutional Neural Networks: Architecture and Theoretical Analysis
• Understanding convolution on graphs via energies
• Graph-Coupled Oscillator Networks
• Graph-Time Convolutional Neural Networks
• Product Graph Learning from Multi-domain Data with Sparsity and Rank\n Constraints
• DiffusionNet: Discretization Agnostic Learning on Surfaces
• DiffusionNet: Discretization Agnostic Learning on Surfaces
• Multi-Graph Tensor Networks
• A Note on Over-Smoothing for Graph Neural Networks
• Continuous Graph Neural Networks
• Graph WaveNet for Deep Spatial-Temporal Graph Modeling
• Graph Signal Processing: Overview, Challenges and Applications
• Geometric Matrix Completion with Recurrent Multi-Graph Neural Networks
• Semi-Supervised Classification with Graph Convolutional Networks
• Tensor Decomposition for Signal Processing and Machine Learning
• Tensor Decompositions and Applications
• A Comprehensive Survey on Graph Neural Networks

Related

• Continuous Product Graph Neural Networks
• Graph Neural Ordinary Differential Equations
• P-Tensors: a General Formalism for Constructing Higher Order Message Passing Networks
• PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations
• DuSEGO: Dual Second-order Equivariant Graph Ordinary Differential Equation