Capsule graph networks for accurate and interpretable crystalline materials property prediction.

Accurate and interpretable modeling of crystalline materials is essential for understanding the structure-property relationships in materials critical in accelerating materials discovery. While recent graph neural networks (GNNs) have achieved high predictive accuracy, they often struggle to provide physical interpretability and fail to explicitly model the hierarchical and symmetrical nature of crystals. In this work, we introduce Capsule Graph Networks with E(3)-Equivariance (CGN-e3), a novel deep learning framework that integrates equivariant message passing with capsule networks to capture both geometric symmetries and hierarchical motif structures. CGN-e3 leverages E(3)-equivariant message passing to learn physically consistent features and organize them into capsule representations that can disentangle local motifs, such as polyhedral environments, and connects them to global properties. We validate the effectiveness of our framework on bandgap and formation energy prediction, as well as material classification using Materials Project and Matbench datasets. Our model achieves a MAE of 0.054 eV/atom and 0.379 eV on formation energy and bandgap prediction, respectively, outperforming CGCNN and matching the performance of MEGNet on the same dataset, while also providing insightful interpretations of the learned capsule representations.Scientific contribution: We present the first integration of E(3)-equivariant graph neural networks with capsule networks for modeling crystalline materials. This unified architecture captures both the fundamental physical symmetries of 3D space; rotation, translation, reflection and the intrinsic hierarchical part-whole relationships e.g., atoms to polyhedra to extended motifs found in crystal structures. The framework provides an unsupervised pathway for interpretable motif discovery. The dynamic routing-by-agreement mechanism identifies and aggregates structurally significant local environments such as the