Equivariant Cellular Sheaves for Molecular Electronic Structure: Bridging Sheaf Cohomology and E(3)-Equivariant Hamiltonian Learning
Equivariant Cellular Sheaves for Molecular Electronic Structure: Bridging Sheaf Cohomology and E(3)-Equivariant Hamiltonian Learning
用于分子电子结构的等变胞腔层：连接层上同调与 E(3)-等变哈密顿量学习
Abstract: Equivariant message-passing networks are the standard model for molecular property and interatomic-potential prediction, and recent work predicts the electronic Hamiltonian itself in an E(3)-equivariant way. Separately, topological deep learning has extended graph networks to cellular sheaves. Our central observation is structural: in a localized atomic-orbital basis, the molecular single-particle Hamiltonian, after a constant shift that makes it positive semidefinite, is the Laplacian of a cellular sheaf on a regular cell complex built from the molecule.
摘要： 等变消息传递网络是分子性质和原子间势能预测的标准模型，近期研究已开始以 E(3)-等变方式预测电子哈密顿量本身。与此同时，拓扑深度学习已将图网络扩展至胞腔层（cellular sheaves）。我们的核心观察是结构性的：在局部原子轨道基组中，分子单粒子哈密顿量在经过使其半正定的常数平移后，本质上是构建于分子之上的正则胞腔复形上的胞腔层拉普拉斯算子。
Making the restriction maps O(3)-steerable two-center kernels from bond geometry recovers the Slater-Koster form as a special case and yields an E(3)- and permutation-equivariant operator. Three consequences follow. First, the zeroth sheaf cohomology H^0 = ker L is a topological invariant equal to the non-bonding (zero-mode) orbitals, recovering the classical alternant non-bonding-orbital count as a lower bound.
通过将限制映射设为基于键几何的 O(3)-可操纵双中心核，我们不仅将 Slater-Koster 形式作为特例复现，还得到了一个 E(3)-等变且置换等变的算子。这带来了三个结论：首先，零阶层上同调 H^0 = ker L 是一个拓扑不变量，等同于非键（零模）轨道，从而将经典的交替非键轨道计数作为其下界。
Second, the Hodge 1-Laplacian lets higher cells (rings) carry cycle and delocalization information through H^1. Third, the model strictly generalizes E(3)-equivariant message-passing networks and CW networks, and inherits the anti-oversmoothing of non-trivial sheaf diffusion.
其次，Hodge 1-拉普拉斯算子使得高阶胞腔（环）能够通过 H^1 携带环路和离域信息。第三，该模型严格推广了 E(3)-等变消息传递网络和 CW 网络，并继承了非平凡层扩散的抗过平滑特性。
We prove equivariance, expressivity, and cohomological-correspondence results for the Equivariant Cellular Sheaf Networks, and validate them numerically: the Hamiltonian-to-sheaf embedding is exact to machine precision, the cohomology dimension reproduces non-bonding-orbital counts across eleven conjugated molecules, the sheaf Laplacian is O(3)-equivariant to machine precision, and the equivariant model attains lower error and rotation generalization on a directional electronic target. Our contribution is this sheaf-theoretic formalization and its invariants, not equivariant Hamiltonian prediction itself.
我们证明了等变胞腔层网络的等变性、表达能力和上同调对应结果，并进行了数值验证：哈密顿量到层的嵌入在机器精度下是精确的；上同调维度在十一种共轭分子中准确复现了非键轨道计数；层拉普拉斯算子在机器精度下满足 O(3)-等变性；且该等变模型在定向电子目标上实现了更低的误差和更好的旋转泛化能力。我们的贡献在于这种层论形式化及其不变量，而非等变哈密顿量预测本身。