Hypergraph Neural Diffusion: A PDE-Inspired Framework for Hypergraph Message Passing

2026-04-12

AI Summary

Existing hypergraph neural networks are limited by shallow propagation, oversmoothing, and inadequate adaptability to complex structures. This work proposes a novel approach that, for the first time, introduces nonlinear partial differential equations (PDEs) into hypergraph learning by formulating a feature propagation mechanism based on a continuous-time anisotropic diffusion equation. The diffusion process is governed by local inconsistency measures and learnable structure-aware diffusion coefficients, with message passing implemented via discrete gradient flows. This framework offers a physically interpretable perspective and enables the construction of deep, stable, and explainable architectures, backed by theoretical guarantees including energy dissipation, boundedness of solutions, and numerical stability. Extensive experiments on multiple benchmark datasets demonstrate superior performance, validating the effectiveness of PDE-driven design in enhancing the expressiveness, stability, and interpretability of hypergraph learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Hypergraph neural networks (HGNNs) have shown remarkable potential in modeling high-order relationships that naturally arise in many real-world data domains. However, existing HGNNs often suffer from shallow propagation, oversmoothing, and limited adaptability to complex hypergraph structures. In this paper, we propose Hypergraph Neural Diffusion (HND), a novel framework that unifies nonlinear diffusion equations with neural message passing on hypergraphs. HND is grounded in a continuous-time hypergraph diffusion equation, formulated via hypergraph gradient and divergence operators, and modulated by a learnable, structure-aware coefficient matrix over hyperedge-node pairs. This partial differential equation (PDE) based formulation provides a physically interpretable view of hypergraph learning, where feature propagation is understood as an anisotropic diffusion process governed by local inconsistency and adaptive diffusion coefficient. From this perspective, neural message passing becomes a discretized gradient flow that progressively minimizes a diffusion energy functional. We derive rigorous theoretical guarantees, including energy dissipation, solution boundedness via a discrete maximum principle, and stability under explicit and implicit numerical schemes. The HND framework supports a variety of integration strategies such as non-adaptive-step (like Runge-Kutta) and adaptive-step solvers, enabling the construction of deep, stable, and interpretable architectures. Extensive experiments on benchmark datasets demonstrate that HND achieves competitive performance. Our results highlight the power of PDE-inspired design in enhancing the stability, expressivity, and interpretability of hypergraph learning.

Problem

• hypergraph neural networks
• oversmoothing
• shallow propagation
• structural adaptability
• high-order relationships

Innovation

• hypergraph neural networks
• neural diffusion
• partial differential equations
• anisotropic diffusion
• message passing