We present both a theoretical and a methodological framework that addresses a
critical challenge in applying deep learning to physical systems: the
reconciliation of non-linear expressiveness with SO(3)-equivariance in
predictions of SO(3)-equivariant quantities, such as the electronic-structure
Hamiltonian. Inspired by covariant theory in physics, we address this problem
by exploring the mathematical relationships between SO(3)-invariant and
SO(3)-equivariant quantities and their representations. We first construct
theoretical SO(3)-invariant quantities derived from the SO(3)-equivariant
regression targets, and use these invariant quantities as supervisory labels to
guide the learning of high-quality SO(3)-invariant features. Given that
SO(3)-invariance is preserved under non-linear operations, the encoding process
for invariant features can extensively utilize non-linear mappings, thereby
fully capturing the non-linear patterns inherent in physical systems. Building
on this foundation, we propose a gradient-based mechanism to induce
SO(3)-equivariant encodings of various degrees from the learned SO(3)-invariant
features. This mechanism can incorporate non-linear expressive capabilities
into SO(3)-equivariant representations, while theoretically preserving their
equivariant properties as we prove. Our approach offers a promising general
solution to the critical dilemma between equivariance and non-linear
expressiveness in deep learning methodologies. We apply our theory and method
to the electronic-structure Hamiltonian prediction tasks, demonstrating
state-of-the-art performance across six benchmark databases.

我们提出了一种理论和方法论框架，以解决在将深度学习应用于物理系统时的一个关键挑战：在预测 SO (3)- 等变量（如电子结构哈密顿量）时，非线性表达能力与 SO (3)- 等变性的协调问题。在物理学中协变理论的启发下，我们通过探索 SO (3)- 不变量和 SO (3)- 等变量及其表示之间的数学关系来解决这个问题。我们首先构造了从 SO (3)- 等变回归目标得出的理论 SO (3)- 不变量，并将这些不变量作为监督标签来指导学习高质量的 SO (3)- 不变特征。由于非线性操作能保持 SO (3)- 不变性，因此不变特征的编码过程可以广泛利用非线性映射，从而充分捕捉物理系统中固有的非线性模式。在此基础上，我们提出了一种基于梯度的机制，从学习到的 SO (3)- 不变特征中诱导出各种程度的 SO (3)- 等变编码。该机制能将非线性表达能力合并到 SO (3)- 等变表示中，并在理论上保持它们的等变性质，我们通过数学证明了这一点。我们的方法为解决深度学习方法中等变性和非线性表达能力之间的关键困境提供了一个有前景的通用解决方案。我们将理论和方法应用于电子结构哈密顿量的预测任务，并在六个基准数据库上展示了最先进的性能。