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

Juwei Yue, Haikuo Li, Jiawei Sheng, Xiaodong Li, Taoyu Su, Tingwen Liu, Li Guo

TL;DR

The paper proposes a novel Hyperbolic-PDE GNN that formulates message passing as a system of hyperbolic partial differential equations on graphs, anchoring node representations to the graph’s Laplacian eigenbasis via $\partial^2 \mathbf{X}/\partial t^2 = a^2 \widehat{\mathbf{L}} \mathbf{X}$ and its eigen-decomposition $\widehat{\mathbf{L}}\,\widehat{\mathbf{u}}_i = \widehat{\lambda}_i \widehat{\mathbf{u}}_i$. To make the approach practical, it introduces polynomial approximations in $\mathbf{L}$, replacing expensive eigen-decompositions with a flexible basis of orthogonal polynomials (e.g., Chebyshev), yielding a forward-Euler time-stepping scheme for node embeddings. The key contributions are (i) a theoretical demonstration that the solution space is spanned by Laplacian-derived eigenvectors, (ii) a polynomial-based mechanism to approximate the hyperbolic PDE solution space and enhance nonlinearity handling, and (iii) extensive experiments showing improved performance and interpretable topology-aware features across graph tasks and image-based graph signals. This framework tightly couples wave-like propagation dynamics with spectral graph structure, offering improved expressiveness over traditional MP-based GNNs and a clear bridge to spectral GNNs with practical polynomial filters.

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.

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

TL;DR

The paper proposes a novel Hyperbolic-PDE GNN that formulates message passing as a system of hyperbolic partial differential equations on graphs, anchoring node representations to the graph’s Laplacian eigenbasis via and its eigen-decomposition . To make the approach practical, it introduces polynomial approximations in , replacing expensive eigen-decompositions with a flexible basis of orthogonal polynomials (e.g., Chebyshev), yielding a forward-Euler time-stepping scheme for node embeddings. The key contributions are (i) a theoretical demonstration that the solution space is spanned by Laplacian-derived eigenvectors, (ii) a polynomial-based mechanism to approximate the hyperbolic PDE solution space and enhance nonlinearity handling, and (iii) extensive experiments showing improved performance and interpretable topology-aware features across graph tasks and image-based graph signals. This framework tightly couples wave-like propagation dynamics with spectral graph structure, offering improved expressiveness over traditional MP-based GNNs and a clear bridge to spectral GNNs with practical polynomial filters.

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.

Paper Structure

This paper contains 35 sections, 5 theorems, 52 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations of Graph
  4. Spectral Graph Neural Networks
  5. Hyperbolic Partial Differential Equation
  6. Methodology
  7. A System of Hyperbolic PDEs on Graph
  8. Derivation of Solution Space
  9. Polynomial Approximation of Solution
  10. Hyperbolic-PDE GNNs
  11. Experiments
  12. Node Classification on Graphs
  13. Filter analysis.
  14. Signal Filtering on Images
  15. Filter analysis.
  16. ...and 20 more sections

Key Result

Theorem 3.1

For given system of differential equations (eq:hpde_graph) and a graph signal on any dimension $l$, there exists a solution space determined by a fundamental matrix of solution: Therefore, a graph signal on any dimension $l$ can be expressed as a linear combination of $2n$ linearly independent column vectors $\boldsymbol{\varphi}_1 (t), \boldsymbol{\varphi}_2 (t), \dots, \boldsymbol{\varphi}_{2n}

Figures (2)

  • Figure 1: The graph spectral filter learned by the methods on graph datasets.
  • Figure 2: The image signals processed by the real filter (first row), Hyperbolic-GCN (second row), and GCN (third row).

Theorems & Definitions (10)

  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • proof