arXiv:2409.15600cs.LGphysics.comp-ph2024-09被引 2

提出可区分手性分子的原子系统表示方法,兼具对称性不变与唯一性。

Polyatomic Complexes: A topologically-informed learning representation for atomistic systems

  • 基于O(3)群符号特征分级与多对称求和,构建拓扑感知的几何映射
  • 新方法在多数相互作用构型下能分离对映体,且实现全局注入性
  • 适用于需区分手性的材料与分子建模,如药物设计与催化研究

分子或材料的描述符应满足物理对称性不变、唯一性、连续性、高效性和泛化性。然而这些性质难以同时满足:在O(3)全正交群下不变的描述符无法区分对映体。本文通过引入O(3)的符号特征分级与多对称求和,构造出映射Φ:其偶部通过格拉姆矩阵,可证明为手性盲;奇部由带符号三重积构成,在交互构型的开稠密满测集上可分离对映体。标准唯一性障碍(输出长度固定、分量池化)源于池化规则,通过阶数不超过N的多对称幂和消除。构建的完整描述符Φ⋆基于距离矩阵、带符号体积与原子类型,在所有构型下关于SE(3)×S_N注入。Φ⋆为非构造性,实际使用的是有截断的Φ,阶数为2,运行时间复杂度为O(N²),邻域列表下可达O(N)。代数核心经Lean 4形式验证。因基础对象为胞腔复形,还生成不变且稳定的拓扑特征Ψ(持久同调与霍奇拉普拉斯谱),编码全局环与笼状结构,常规截断描述符无法捕捉。组合(Φ,Ψ)输入紧凑的符号分级等变变压器。

原文摘要 · Abstract (English)

A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general. These properties, however, are hard to satisfy at once: a descriptor invariant under the full orthogonal group $O(3)$ gives a molecule and its mirror image the same value, and so cannot distinguish enantiomers whose properties differ. Pozdnyakov showed this follows from the invariance itself, not from a lack of parameters. We show the criteria can be met at once if the geometric map is graded by the sign character of $O(3)$ and pooled multisymmetrically. We construct such a map $Φ$: its even block factors through the Gram matrix and is provably chirality-blind, while its parity-odd block of signed triple products separates enantiomers on an open dense full-measure set of interacting configurations. Two standard obstructions to uniqueness, fixed output length and componentwise pooling, are artifacts of the pooling rule, removed by multisymmetric power sums of order at most $N$. We establish uniqueness for a complete descriptor $Φ^\star$ built from the distance matrix, signed volumes and atom types, injective up to $SE(3)\times S_N$ on all configurations. $Φ^\star$ is non-constructive, however; the implemented map is the bounded-cutoff $Φ$, generically injective, pooling at order $2$, running in $O(N^2)$, or $O(N)$ with neighbor lists. The algebraic core is machine-checked in Lean 4. Because the underlying object is a cell complex, it also yields invariant, stable topological features $Ψ$ (persistent homology and a Hodge-Laplacian spectrum) encoding global ring and cage structure invisible to bounded-cutoff descriptors. The pair $(Φ,Ψ)$ feeds a compact parity-graded equivariant transformer.

分子表示手性识别拓扑特征等变网络

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