Turbocharging Graph Neural Networks with Zentropy

Graph neural networks (GNNs) learn by aggregating feature representations across connected nodes, a mechanism that excels when graph topology aligns with label consensus. In heterophilic graphs, however, the homophily inductive bias of GNNs often yields suboptimal predictions, as the learning dynamics overemphasize relational smoothness. The popular driving force of GNN is cross-entropy, an objective that minimizes information uncertainty via a zero-temperature energy descent. Because cross-entropy does not account for internal disparities during feature aggregation, current GNN architectures remain trapped in rigid energy regimes that cannot dynamically adapt to node-level structural divergence. To address this limitation, we reformulate GNN optimization through zentropy, a multi-scale thermodynamic framework under a free-energy landscape that explicitly balances internal energy (the likelihood associated with downstream tasks) with intrinsic disparities (a prior on data heterogeneity). In the spirit of thermodynamic principle, we devise a physics-informed neural network to discover the critical temperature for each graph node, creating a new degree of freedom that mitigates representational collapse caused by feature over-smoothing in GNNs. Together, we present a model-agnostic zentropy loss function in the regime of Helmholtz free energy that serves as a plug-and-play module compatible with standard GNN architectures. We evaluate the zentropy-based readout against matched cross-entropy baselines on seven encoders, three citation networks, five heterophilic benchmarks, three OGB graphs and five graph-classification datasets. Our proposed zentropy loss rescues deep encoders in GNN where standard cross-entropy optimization collapses to chance-level performance. Across all evaluated combinations of backbone architectures and network configurations, zentropy loss yields promising performance gains, most notably on the heterophilic graph datasets.