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：在材料研究中，针对量子系统的深度学习哈密顿量回归需满足协方差定律，其中在不牺牲网络表达能力的情况下实现SO(3)-等变性仍是悬而未决的挑战——这是由于在保证理论等变性时对非线性映射的限制所致。为缓解协方差-表达能力的困境，我们提出了一种包含两级级联回归的混合框架。第一阶段采用理论保证的协变神经网络建模三维原子系统的对称性，生成理论协变特征与基线哈密顿量预测，协助第二阶段学习协方差。同时，第二阶段基于我们提出的面向三维原子系统结构建模的非线性3D图Transformer网络，将第一阶段的输出精化为具有更强表达能力的细粒度哈密顿量预测。通过将理论协变但表达能力受限的模型与高表达能力的非线性网络相结合，该方法在保持坐标变换鲁棒协方差的同时，实现了精确且可泛化的预测。在五个晶体材料数据集上的实验证实，我们的方法在电子结构计算的哈密顿量预测中达到了最先进性能。