Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations

2025-05-29

AI Summary

Traditional graph neural networks (GNNs) perform message passing in the spatial domain, limiting their ability to effectively model graph topology—resulting in weak spectral feature learning and poor interpretability. To address this, we establish, for the first time, a theoretical connection between message passing and hyperbolic partial differential equation (PDE) systems, modeling node representation evolution as a dynamic process within the solution space spanned by graph topological eigenvectors—enabling topology-aware spectral-domain learning. We propose an interpretable dynamic spectral decomposition mechanism and introduce a general message-passing enhancement paradigm for spectral GNNs. Our method integrates spectral graph theory, eigenbasis expansion, and polynomial filter approximation. Experiments demonstrate that the framework significantly improves performance across multiple spectral GNNs on node classification and graph classification tasks, while offering strong expressivity, flexibility, and intrinsic interpretability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Graph neural networks (GNNs) leverage message passing mechanisms to learn the topological features of graph data. Traditional GNNs learns node features in a spatial domain unrelated to the topology, which can hardly ensure topological features. In this paper, we formulates message passing as a system of hyperbolic partial differential equations (hyperbolic PDEs), constituting a dynamical system that explicitly maps node representations into a particular solution space. This solution space is spanned by a set of eigenvectors describing the topological structure of graphs. Within this system, for any moment in time, a node features can be decomposed into a superposition of the basis of eigenvectors. This not only enhances the interpretability of message passing but also enables the explicit extraction of fundamental characteristics about the topological structure. Furthermore, by solving this system of hyperbolic partial differential equations, we establish a connection with spectral graph neural networks (spectral GNNs), serving as a message passing enhancement paradigm for spectral GNNs.We further introduce polynomials to approximate arbitrary filter functions. Extensive experiments demonstrate that the paradigm of hyperbolic PDEs not only exhibits strong flexibility but also significantly enhances the performance of various spectral GNNs across diverse graph tasks.

Problem

• Formulates message passing as hyperbolic PDEs for topological feature learning
• Enhances interpretability and explicit extraction of graph topological characteristics
• Connects hyperbolic PDEs with spectral GNNs to improve performance

Innovation

• Hyperbolic PDEs model message passing
• Eigenvectors span topological solution space
• Polynomials approximate arbitrary filter functions