论文标题
用于拓扑数据分析的流量轨道结构的离散表示
Discrete representations of orbit structures of flows for topological data analysis
论文作者
论文摘要
本文表明，由一类矢量场在球形表面上产生的粒子轨道的拓扑结构，称为{\ it有限类型的流}，与离散的结构（例如树/图和字母序列）一对一地对应。有限类型的流量是结构稳定的哈密顿载体场的扩展，这些载体磁场出现在2D不可压缩流体流的许多理论和数值研究中。此外，它包含可压缩的2D矢量场，例如莫尔斯 - 摩尔矢量场以及3D矢量场在2D部分上的投影。离散表示不仅是复杂流拓扑结构的简单符号标识符，而且当应用于测量，实验和复杂流的数值模拟的数据时，它也会引起对流的拓扑数据分析的新方法。作为概念的证明，我们将表示理论的某些应用提供给2D可压缩矢量场和在工业问题中产生的3D矢量场。
This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as trees/graphs and sequence of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of 2D incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse--Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem.
