DuSEGO: Dual Second-Order Equivariant Graph Ordinary Differential Equation

Graph Neural Networks (GNNs) with equivariant properties have achieved significant success in modeling complex dynamic systems and molecular properties. However, their expressiveness ability is limited by: (1) Existing methods often overlook the over-smoothing issue caused by traditional GNN models, as well as the gradient explosion or vanishing problems in deep GNNs. (2) Most models operate on first-order information, neglecting that the real world often consists of second-order systems, which further limits the model’s representation capabilities. To address these issues, we propose the Dual Second-order Equivariant Graph Ordinary (DuSEGO) Differential Equation for equivariant representation. Specifically, DuSEGO applies the dual second-order equivariant graph ordinary differential equations (Graph ODEs) to both graph embeddings and node coordinates simultaneously. Theoretically, we first prove that DuSEGO maintains the equivariant property. Furthermore, we provide theoretical insights showing that DuSEGO effectively alleviates the over-smoothing problem in both feature representation and coordinate update. Additionally, we demonstrate that the proposed DuSEGO mitigates the exploding and vanishing gradients problem, facilitating the training of deep multi-layer GNNs. Extensive experiments on benchmark datasets validate the superiority of the proposed DuSEGO compared to baselines.