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.

深度学习用于材料研究中的哈密顿回归需要满足协方差定律，其中在保证网络表达能力的前提下实现 SO (3) 等变性仍然是个困难的挑战。为了解决协方差 - 表达能力两难问题，我们提出了一个混合框架，利用两个级联回归阶段。第一个阶段，利用一个理论上保证协变的神经网络对 3D 原子系统的对称性建模，产生理论上协变的特征和基线哈密顿预测，辅助第二阶段在学习协变性方面。与此同时，第二阶段由我们提出的非线性 3D 图 Transformer 网络驱动，用于对 3D 原子系统进行结构建模，将第一阶段的输出细化为具有更好表达能力的哈密顿精细预测。理论上协变但不可避免地表达能力较差的模型与高度表达能力的非线性网络的组合，能够实现精确、可推广的预测，并在坐标变换下保持稳健的协方差。我们的方法在电子结构计算的哈密顿预测方面达到了最先进的性能，在五个晶体材料数据库上进行的实验证实了这一点。

混合串联回归框架的晶态材料研究中调和协方差性和表达性的深哈密顿回归

Harmonizing Covariance and Expressiveness for Deep Hamiltonian  Regression in Crystalline Material Research: a Hybrid Cascaded Regression  Framework

Recent advances in computational modelling of atomic systems, spanning
molecules, proteins, and materials, represent them as geometric graphs with
atoms embedded as nodes in 3D Euclidean space. In these graphs, the geometric
attributes transform according to the inherent physical symmetries of 3D atomic
systems, including rotations and translations in Euclidean space, as well as
node permutations. In recent years, Geometric Graph Neural Networks have
emerged as the preferred machine learning architecture powering applications
ranging from protein structure prediction to molecular simulations and material
generation. Their specificity lies in the inductive biases they leverage --
such as physical symmetries and chemical properties -- to learn informative
representations of these geometric graphs. In this opinionated paper, we
provide a comprehensive and self-contained overview of the field of Geometric
GNNs for 3D atomic systems. We cover fundamental background material and
introduce a pedagogical taxonomy of Geometric GNN architectures:(1) invariant
networks, (2) equivariant networks in Cartesian basis, (3) equivariant networks
in spherical basis, and (4) unconstrained networks. Additionally, we outline
key datasets and application areas and suggest future research directions. The
objective of this work is to present a structured perspective on the field,
making it accessible to newcomers and aiding practitioners in gaining an
intuition for its mathematical abstractions.

计算模型，几何图，几何图神经网络，三维原子系统，机器学习架构