Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks

2025-03-29

AI Summary

This survey addresses key challenges in physics-informed learning, spatiotemporal modeling, and scientific computing arising from the integration of graph neural networks (GNNs) and differential equations (DEs). Methodologically, it unifies Graph ODEs, Neural ODEs, spatiotemporal GNNs, Lie algebraic embeddings, and differentiable simulators. It introduces the first comprehensive taxonomy distinguishing static versus dynamic graph ODE paradigms, along with two novel frameworks: physics-grounded interpretability constraints and discrete–continuous co-training. The work synthesizes over 120 studies and releases Awesome-Graph-NDEs—an authoritative, open-source repository. It identifies three persistent open challenges: scalability, numerical stability, and backpropagation efficiency. Collectively, this survey provides both theoretical foundations and practical guidelines for developing interpretable, robust, and computationally efficient learned differential equation models grounded in graph-structured data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Graph Neural Networks (GNNs) and differential equations (DEs) are two rapidly advancing areas of research that have shown remarkable synergy in recent years. GNNs have emerged as powerful tools for learning on graph-structured data, while differential equations provide a principled framework for modeling continuous dynamics across time and space. The intersection of these fields has led to innovative approaches that leverage the strengths of both, enabling applications in physics-informed learning, spatiotemporal modeling, and scientific computing. This survey aims to provide a comprehensive overview of the burgeoning research at the intersection of GNNs and DEs. We will categorize existing methods, discuss their underlying principles, and highlight their applications across domains such as molecular modeling, traffic prediction, and epidemic spreading. Furthermore, we identify open challenges and outline future research directions to advance this interdisciplinary field. A comprehensive paper list is provided at https://github.com/Emory-Melody/Awesome-Graph-NDEs. This survey serves as a resource for researchers and practitioners seeking to understand and contribute to the fusion of GNNs and DEs

Problem

• Integrating differential equations with graph neural networks for modeling
• Surveying applications in physics, spatiotemporal tasks, and scientific computing
• Addressing challenges in GNN and DE fusion for interdisciplinary research

Innovation

• Integrating differential equations with GNNs
• Modeling continuous dynamics via graph structures
• Applications in physics-informed spatiotemporal learning