Applications of Geometric and Group-theoretic Methods to Network Dynamics

科学中的一个根本挑战是理解复杂系统中集体行为是如何出现的。例如，组成心脏自然起搏器网络的数千个心脏细胞如何同时发出电脉冲？一群鸟或一群鱼是如何组织在一起飞行或游泳的？是什么导致一群萤火虫同时闪烁？我们如何防止大型电网中的单个组件故障级联到整个网络？研究人员对振荡器网络特别感兴趣，在这个网络中，各个组件在时间上周期性地循环。振荡器网络可以表现出各种集体行为，简单地说，从所有振荡器一致行为时的完全同步，到更复杂的模式，如嵌合体状态，其中网络的一部分同步，而其余部分表现出异步行为。研究人员开发了数学工具，使用几何、群论和降维，极大地简化了振荡器网络及其稳定性的分析。这种方法的优点是，只使用少量的方程就可以精确地描述具有许多变量的大型网络的行为。他们使用这些方法来设计和分析网络，并将其应用于油藏计算和机器学习，这是现代数据分析的一个重要和发展中的领域。这项工作的一个重要分支是在振子动力学更实用的研究和低维几何的纯数学研究之间建立跨学科的合作。研究生从事该项目的研究。该项目的中心主题是应用几何和群论技术来研究Kuramoto网络和更一般类型的振子。几何技术的应用使得显式描述群轨道上的动力学成为可能。在使用这种方法的早期工作的基础上，调查人员提出了四个主题。第一种方法是对具有几何(梯度)结构的高阶参数函数进行分类，这种几何(梯度)结构类似于具有一阶参数的经典Kuramoto模型。主题2使用几何技术来了解多种群模型的动力学，特别是阐明嵌合体状态的动力学，嵌合体状态是一些但不是所有种群完全同步的状态。主题3将这种结构扩展到高维振荡器的网络，这些振荡器的状态在2维或更高的球体上，用于对群、群和其他社会网络进行建模。主题4将该技术应用于机器学习和水库计算。研究生从事该项目的研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A fundamental challenge in science is to understand how collective behavior emerges in complex systems. For example, how do the thousands of heart cells that form the heart's natural pacemaker network manage to fire an electrical pulse simultaneously? How does a flock of birds or school of fish organize so as to fly or swim together? What causes a population of fireflies to flash in unison? How can we prevent a single component failure in a large power grid from cascading through the entire network? The investigators' particular interest is in networks of oscillators, in which the individual components cycle periodically in time. Oscillator networks can exhibit a variety of collective behaviors, ranging in simplicity from complete synchronization, when all the oscillators behave in unison, to more complex patterns such as chimera states, where part of the network is synchronized while the rest exhibits asynchronous behavior. The investigators develop mathematical tools, using geometry, group theory, and dimensional reduction, to greatly simplify the analysis of oscillator networks and their stability. The virtue of this approach is that the behavior of a large network with many variables can be precisely described using only a small number of equations. They use these methods to design and analyze networks with applications to reservoir computing and machine learning, an important and developing area of modern data analysis. An important ramification of this work is to forge interdisciplinary collaborations between the more applied study of oscillator dynamics and the pure mathematical study of low-dimensional geometry. Graduate students are engaged in the research of the project.The central theme of this project is the application of geometric and group-theoretic techniques to the study of networks of Kuramoto and more general types of oscillators. The application of geometric techniques makes possible the explicit description of the dynamics on group orbits. Building on on their earlier work using this methodology, the investigators take up four topics. The first classifies higher degree order parameter functions that have a geometric (gradient) structure similar to the classic Kuramoto model with first degree order parameter. Topic 2 uses geometric techniques to understand the dynamics of multi-population models, particularly to clarify the dynamics of chimera states, which are states with some but not all of the populations completely synchronized. Topic 3 extends this structure to networks of higher-dimensional oscillators, with states on spheres of dimension 2 or higher, that are used to model flocks, swarms and other social networks. Topic 4 applies this technology to machine learning and reservoir computing. Graduate students are engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.