Dirichlet Energy

Dirichlet energy, a measure of the smoothness of a function on a graph, is central to addressing the "over-smoothing" problem in graph neural networks (GNNs), where deep networks lose discriminative power. Current research focuses on modifying GNN architectures, such as incorporating framelets or fractional graph Laplacians, to control and enhance Dirichlet energy during training, thereby improving performance, particularly on heterophilous graphs. This work also extends to applications beyond GNNs, including feature selection, implicit neural representations, and shape matching, highlighting the broad utility of Dirichlet energy as a regularizer for improving model generalization and performance.