Pozdnyakov Sergey N, Zhang Liwei, Ortner Christoph, Csányi Gábor, Ceriotti Michele
Laboratory of Computational Science and Modelling, Institute of Materials, Federal Institute of Technology (EPFL) CH-1015 Lausanne, Lausanne, 1015, Switzerland.
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada.
Open Res Eur. 2021 Oct 20;1:126. doi: 10.12688/openreseurope.14156.1. eCollection 2021.
The increasingly common applications of machine-learning schemes to atomic-scale simulations have triggered efforts to better understand the mathematical properties of the mapping between the Cartesian coordinates of the atoms and the variety of representations that can be used to convert them into a finite set of symmetric or . Here, we analyze the sensitivity of the mapping to atomic displacements, using a singular value decomposition of the Jacobian of the transformation to quantify the sensitivity for different configurations, choice of representations and implementation details. We show that the combination of symmetry and smoothness leads to mappings that have singular points at which the Jacobian has one or more null singular values (besides those corresponding to infinitesimal translations and rotations). This is in fact desirable, because it enforces physical symmetry constraints on the values predicted by regression models constructed using such representations. However, besides these symmetry-induced singularities, there are also spurious singular points, that we find to be linked to the of the mapping, i.e. the fact that, for certain classes of representations, structurally distinct configurations are not guaranteed to be mapped onto different feature vectors. Additional singularities can be introduced by a too aggressive truncation of the infinite basis set that is used to discretize the representations. These results exemplify the subtle issues that arise when constructing symmetric representations of atomic structures, and provide conceptual and numerical tools to identify and investigate them in both benchmark and realistic applications.
机器学习方法在原子尺度模拟中的应用日益普遍，这引发了人们对更好地理解原子笛卡尔坐标与可用于将其转换为有限对称集或其他表示形式之间映射的数学性质的努力。在此，我们使用变换雅可比矩阵的奇异值分解来量化不同构型、表示选择和实现细节下映射对原子位移的敏感性，从而分析该映射的敏感性。我们表明，对称性和平滑性的结合导致映射存在奇点，在这些奇点处雅可比矩阵具有一个或多个零奇异值（除了对应于无穷小平移和旋转的那些）。事实上，这是可取的，因为它对使用此类表示构建的回归模型预测的值施加了物理对称性约束。然而，除了这些由对称性引起的奇点之外，还存在虚假奇点，我们发现这些虚假奇点与映射的退化有关，即对于某些类别的表示，结构上不同的构型不能保证映射到不同的特征向量上。过度激进地截断用于离散化表示的无限基集可能会引入额外的奇点。这些结果例证了在构建原子结构的对称表示时出现的微妙问题，并提供了概念和数值工具，以便在基准和实际应用中识别和研究这些问题。
