Graph Neural Networks as Gradient Flows

Dynamical systems minimizing an energy are ubiquitous in geometry and physics. We propose a gradient flow framework for GNNs where the equations follow the direction of steepest descent of a learnable energy. This approach allows to explain the GNN evolution from a multi-particle perspective as learning attractive and repulsive forces in feature space via the positive and negative eigenvalues of a symmetric "channel-mixing" matrix. We perform spectral analysis of the solutions and conclude that gradient flow graph convolutional models can induce a dynamics dominated by the graph high frequencies which is desirable for heterophilic datasets. We also describe structural constraints on common GNN architectures allowing to interpret them as gradient flows. We perform thorough ablation studies corroborating our theoretical analysis and show competitive performance of simple and lightweight models on real-world homophilic and heterophilic datasets.