Response properties of molecules and crystals are naturally described by tensors that obey specific equivariance and symmetry constraints. However, directly predicting these tensorial quantities remains challenging for machine learning models. We present a general-purpose output module for equivariant graph neural networks that enables end-to-end prediction of tensors of arbitrary order with prescribed permutation (fundamental) symmetry. Coupled with the SE(3)-equivariant XPaiNN architecture, our framework attains accuracy comparable to that of first-principles calculations. It also supports atomic-level properties─such as chemical shielding tensors and Born effective charges─in an all-in-one model. Moreover, the method handles higher-order tensors, including molecular hyperpolarizability and the elastic tensor (stiffness matrix) of crystalline materials, thereby enabling the derivation and analysis of rich anisotropic information and facilitating AI-assisted discovery and design of functional molecules and materials.
Yan et al. (Tue,) studied this question.