发表机构
Elkins High School（埃尔金斯高中）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出等变胞层网络，将分子哈密顿量建模为胞层拉普拉斯算子，兼具拓扑性质与E(3)等变性，在共轭分子非键轨道预测等任务上性能更优。
AI 中文摘要
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，近期研究已实现以E(3)等变方式预测电子哈密顿量。另一方面，拓扑深度学习已将图网络扩展至胞层。本文核心结构观察为：在局域原子轨道基中，分子单粒子哈密顿量经使之为半正定的常数平移后，成为由分子构建的正则胞复形上胞层的拉普拉斯算子。将限制映射设为源自键几何的O(3)可调控双中心核，可在特殊情形下恢复Slater-Koster形式，并得到E(3)和置换等变算子。由此产生三个结论：第一，第0层上同调H^0 = ker L是拓扑不变量，等于非键（零模）轨道，将经典交替非键轨道数作为下界；第二，霍奇1-拉普拉斯算子使高阶胞（环）通过H^1承载环与离域信息；第三，该模型严格泛化E(3)等变消息传递网络和CW网络，并继承非平凡层扩散的抗过平滑特性。本文对等变胞层网络证明了等变性、表达性和上同调对应结果，并进行数值验证：哈密顿量到层的嵌入精确至机器精度，11个共轭分子的上同调维数重现非键轨道数，层拉普拉斯算子O(3)等变至机器精度，且等变模型在定向电子目标上实现更低误差和旋转泛化。本文贡献为该层论形式化及其不变量，而非等变哈密顿预测本身。
英文摘要
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. 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. 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. 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.
Comments12 pages, 3 figures, 2 tables
