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Deep learning for predicting the electronic-structure Hamiltonian of quantumsystems necessitates satisfying the covariance laws, among which achievingSO(3)-equivariance without sacrificing the non-linear expressive capability ofnetworks remains unsolved. To navigate the harmonization between equivarianceand expressiveness, we propose a deep learning method synergizing two distinctcategories of neural mechanisms as a two-stage encoding and regressionframework. The first stage corresponds to group theory-based neural mechanismswith inherent SO(3)-equivariant properties prior to the parameter learningprocess, while the second stage is characterized by a non-linear 3D graphTransformer network we propose, featuring high capability on non-linearexpressiveness. The novel combination lies in the point that, the first stagepredicts baseline Hamiltonians with abundant SO(3)-equivariant featuresextracted, assisting the second stage in empirical learning of equivariance;and in turn, the second stage refines the first stage's output as afine-grained prediction of Hamiltonians using powerful non-linear neuralmappings, compensating for the intrinsic weakness on non-linear expressivenesscapability of mechanisms in the first stage. Our method enables precise,generalizable predictions while capturing SO(3)-equivariance under rotationaltransformations, and achieves state-of-the-art performance in Hamiltonianprediction on six benchmark databases.

Towards Harmonization of SO(3)-Equivariance and Expressiveness: a Hybrid Deep Learning Framework for Electronic-Structure Hamiltonian Prediction