Harmonizing SO(3)-equivariance and Expressiveness for Deep Hamiltonian Regression in Crystalline Material Research

Deep learning for Hamiltonian regression of quantum systems in material research necessitates satisfying the covariance laws, among which achieving SO(3)-equivariance without sacrificing the non-linear expressive capability of networks remains unsolved. To navigate the harmonization between equivariance and expressiveness, we propose a hybrid method synergizing two distinct categories of network mechanisms as a two-stage cascaded regression framework. The first stage corresponds to group theory-based network mechanisms with inherent SO(3)-equivariant properties prior to the parameter learning process, while the second stage is characterized by a non-linear 3D graph Transformer network we propose featuring high capability on non-linear expressiveness. The novel combination lies that, the first stage predicts baseline Hamiltonians with abundant SO(3)-equivariant features extracted, assisting the second stage in empirical learning of equivariance; and in turn, the second stage refines the first stage's output as a fine-grained prediction of Hamiltonians using powerful non-linear neural mappings, compensating for the intrinsic weakness on non-linear expressiveness capability of mechanisms in the first stage. Our method enables precise, generalizable predictions while maintaining robust SO(3)-equivariance under rotational transformations, and achieves state-of-the-art performance in Hamiltonian prediction, confirmed through experiments on six crystalline material databases. Our research provides a new technical pathway for high-performance electronic structure calculations of atomic systems, offering powerful technological means for the simulation, design, and discovery of new materials.