Graph Neural Networks as Gradient Flows

Dynamical systems minimizing an energy are ubiquitous in geometry and physics. We propose a novel framework for GNNs where we parametrize (and {\em learn}) an energy functional and then take the GNN equations to be the gradient flow of such energy. This approach allows to analyse 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 conduct spectral analysis of the solutions and provide a better understanding of the role of the channel-mixing in (residual) graph convolutional models and of its ability to steer the diffusion away from over-smoothing. We perform thorough ablation studies corroborating our theory and show competitive performance of simple models on homophilic and heterophilic datasets.