Graphon Neural Differential Equations and Transferabilty of Graph Neural Differential Equations

Mingsong Yan, Charles Kulick, Sui Tang

Keywords: Graphon Neural Networks, Graphon Neural Differential Equations, Transferabilty, Data-driven Modeling, Generalization, Graph Limits

Abstract: Graph Neural Differential Equations (GNDEs) extend Graph Neural Networks (GNNs) to a continuous-depth framework, providing a robust tool for modeling complex network dynamics. In this paper, we investigate the potential of GNDEs for transferring knowledge across different graphs with shared convolutional structures. To bridge the gap between discrete and continuous graph representations, we introduce Graphon Neural Differential Equations (Graphon-NDEs) as the continuous limit of GNDEs. Using tools from nonlinear evolution equations and graph limit theory, we rigorously establish this continuum limit and develop a mathematical framework to quantify the approximation error between a GNDE and its corresponding Graphon-NDE, which decreases as the number of nodes increases, ensuring reliable transferability. We further derive specific rates for various graph families, providing practical insights into the performance of GNDEs. These findings extend recent results on GNNs to the continuous-depth setting and reveal a fundamental trade-off between discriminability and transferability in GNDEs.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 9190