Physics-Routed MoE with Crystallographic Priors for Long-Tailed Space-Group Recognition

Determining a crystal's space group from a powder X-ray diffraction (PXRD) pattern is a 230-way fine-grained classification problem characterized by severe class imbalance and substantial similarity among diffraction patterns. Many rare space groups have fewer than 20 training examples and are easily confused with frequent groups in the same crystal system. We present Phys-MoE, a Mixture-of-Experts classifier that incorporates crystallographic priors into routing and representation learning. Phys-MoE consists of three components: a hierarchical router that masks space groups inconsistent with the predicted crystal system, a rule-biased gating mechanism that encourages expert specialization across symmetry blocks, and a rule-semantic contrastive objective based on the Jaccard similarity between symmetry-operation sets. Experiments on CCDC, CoREMOF, and an inorganic distribution-shift benchmark demonstrate the effectiveness and generalizability of Phys-MoE. On CCDC, Phys-MoE achieves 81.70% overall Top-1 accuracy and 26.19% accuracy for space groups with fewer than 20 training examples, establishing a new state of the art for long-tailed space-group recognition from PXRD patterns. Ablation studies demonstrate that the three crystallographic components work jointly to improve recognition, while error and oracle analyses identify crystal-system prediction as a remaining bottleneck for rare classes. These results show that incorporating crystallographic structure into expert routing and representation learning provides an effective approach to accurate and reliable space-group recognition under long-tailed data distributions.