Learning Generalized Hessian on Graphs for Continuous Graph Learning

In this work, we propose HbInOut (Hessian-based Influx Outflux), a continuous graph neural network for graph learning and graph ensemble (snapshot) learning. HbInOut models diffusion on graphs by incorporating flow-based priors in its forward-pass construction and includes directionality through learnable influx and outflux operators. We analyze key theoretical properties of the HbInOut diffusion and empirically demonstrate that HbInOut mitigates oversmoothing. Across a diverse set of graph datasets, HbInOut achieves strong performance and exceeds state-of-the-art baselines, with particularly clear gains on directed and flow-structured datasets. For example, HbInOut attains up to 4\% gain in test accuracy on a graph learning task, and reduces 5.5 times test MSE on a synthetic graph ensemble task.