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 expressiveness of networks remains an elusive challenge due to the restriction to non-linear mappings on guaranteeing theoretical equivariance. To alleviate the covariance-expressiveness dilemma, we propose a hybrid framework with two cascaded regression stages. The first stage, with a theoretically-guaranteed covariant neural network modeling symmetry properties of 3D atom systems, yields theoretically covariant features and baseline Hamiltonian predictions, assisting the second stage in learning covariance. Meanwhile, the second stage, powered by a non-linear 3D graph Transformer network we propose for structural modeling of 3D atomic systems, refines the first stage's output as a fine-grained prediction of Hamiltonians with better expressiveness capability. The combination of a theoretically covariant yet inevitably less expressive model with a highly expressive non-linear network enables precise, generalizable predictions while maintaining robust covariance under coordinate transformations. Our method achieves state-of-the-art performance in Hamiltonian prediction for electronic structure calculations, confirmed through experiments on five crystalline material databases.

翻译：材料研究中量子系统哈密顿回归的深度学习需满足协方差律，其中在保证理论协方差的前提下实现SO(3)-等变性而不牺牲网络表达性，因受限于非线性映射的理论约束而成为棘手挑战。为缓解协方差-表达性困境，我们提出包含两级级联回归的混合框架。第一级采用具有理论保证的协变神经网络建模三维原子系统的对称性质，输出理论协变特征与哈密顿基线预测，辅助第二级学习协方差。同时，第二级通过我们提出的用于三维原子系统结构建模的非线性图Transformer网络，将第一级输出精炼为具有更强表达能力的细粒度哈密顿预测。理论协变但表达性受限模型与高表达非线性网络的组合，使得预测在坐标变换下保持稳健协方差的同时，兼具精确性与泛化能力。在五个晶态材料数据库上的实验结果表明，本方法在电子结构计算的哈密顿预测中达到了先进水平。