Material symmetry recognition and property prediction accomplished by crystal capsule representation.

Learning the global crystal symmetry and interpreting the equivariant information is crucial for accurately predicting material properties, yet remains to be fully accomplished by existing algorithms based on convolution networks. To overcome this challenge, here we develop a machine learning (ML) model, named symmetry-enhanced equivariance network (SEN), to build material representation with joint structure-chemical patterns, to encode important clusters embedded in the crystal structure, and to learn pattern equivariance in different scales via capsule transformers. Quantitative analyses of the intermediate matrices demonstrate that the intrinsic crystal symmetries and interactions between clusters have been exactly perceived by the SEN model and critically affect the prediction performances by reducing effective feature space. The mean absolute errors (MAEs) of 0.181 eV and 0.0161 eV/atom are obtained for predicting bandgap and formation energy in the MatBench dataset. The general and interpretable SEN model reveals the potential to design ML models by implicitly encoding feature relationship based on physical mechanisms.