Applications of Geometric and Group-theoretic Methods to Network Dynamics

科学的一个基本挑战是理解复杂系统中的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是如何同时发出电脉冲的？一群鸟或一群鱼是如何组织起来一起飞行或游泳的？是什么导致一群萤火虫一起闪光？我们如何防止大型电网中单个组件的故障在整个网络中发生连锁反应？研究人员特别感兴趣的是振子网络，其中单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各样的集体行为，从简单的完全同步，当所有振荡器的行为一致时，到更复杂的模式，如嵌合体状态，其中部分网络是同步的，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，大大简化了振荡器网络及其稳定性的分析。这种方法的优点是，一个具有许多变量的大型网络的行为可以用少量的方程精确地描述。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且正在发展的领域。这项工作的一个重要分支是在振荡器动力学的更多应用研究和低维几何的纯数学研究之间建立跨学科合作。研究生从事该项目的研究。这个项目的中心主题是几何和群论技术的应用，以研究Kuramoto网络和更一般类型的振荡器。几何技术的应用使群轨道动力学的显式描述成为可能。在他们使用这种方法的早期工作的基础上，研究人员研究了四个主题。第一个分类具有几何（梯度）结构的高次阶参数函数，类似于具有一阶阶参数的经典Kuramoto模型。主题2使用几何技术来理解多种群模型的动力学，特别是澄清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将这种结构扩展到具有2维或更高维度球体状态的高维振子网络，用于模拟羊群、群体和其他社会网络。主题4将该技术应用于机器学习和油藏计算。研究生从事该项目的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.