  In this work, we provide a theoretical understanding of the framelet-based
graph neural networks through the perspective of energy gradient flow. By
viewing the framelet-based models as discretized gradient flows of some energy,
we show it can induce both low-frequency and high-frequency-dominated dynamics,
via the separate weight matrices for different frequency components. This
substantiates its good empirical performance on both homophilic and
heterophilic graphs. We then propose a generalized energy via framelet
decomposition and show its gradient flow leads to a novel graph neural network,
which includes many existing models as special cases. We then explain how the
proposed model generally leads to more flexible dynamics, thus potentially
enhancing the representation power of graph neural networks.
