Data-Driven Learning and Geometric Embedding for Reduction and Control of Complex Heterogeneous Networks

多个主体（组件）动态地相互影响的复杂系统在自然界和人类社会中以不同的尺度普遍存在，例如大脑中的神经元，蜂巢中的蜜蜂和社会网络中的人类。这种系统的不良行为，以疾病，经济崩溃，谣言传播和社会动荡的形式，已经引起了人们对理解这种复杂网络的动态结构和设计控制它们的方法的极大兴趣。尽管数据丰富，易于访问，并且数据科学取得了进步，但获得此类网络的可靠模型仍然是一个非常具有挑战性的问题。这些新兴的复杂系统的规模也带来了很大的困难。这些障碍也形成了用于分析和设计动态结构的瓶颈（例如，同步和集群），并用于控制这种复杂网络中的集体行为。该项目将开发一个统一的数据驱动框架，以研究如何从其仿真或测量数据中提取大规模复杂系统或网络的动力学，以及如何控制该系统，如果动力学重建是成功和可靠的。该项目还将通过创造暑期研究机会，支持新的举措，促进密苏里州圣路易斯市当地高中传统上教育水平低下的学生的跨学科教育，通过将系统和控制理论与代数几何、时间序列分析和机器学习的概念和方法联系起来，将建立一个统一的数据驱动框架。具体而言，一种新的方法，基于谱分解将开发提取一个复杂的系统的动态和解码的复杂网络的拓扑结构使用其时间序列数据。重建网络的属性，例如，连接性和节点的耦合强度，然后将被用来合成一个动态接近的减少网络，这是易于控制理论分析和设计。此外，新的拓扑和几何方法将被导出来构造高维数据到低维流形的局部和全局嵌入，这将揭示隐藏在大数据集中的拓扑结构，并表征底层动力系统的流的转换。在与生物学和化学研究人员的合作中，网络推理，降维和控制技术将应用于从细胞到社会的各种复杂系统，例如，该奖项反映了NSF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。

Complex systems in which multiple agents (components) affect each other dynamically are prevalent in nature and human society in different scales, such as neurons in the brain, bees in a hive, and human beings in a social network. Undesirable behavior of such systems, in the form of disease, economic collapse, rumor spreading, and social unrest, has generated considerable interest in understanding the dynamic structures of such complex networks and devising ways to control them. Despite the abundance of data, ease of access, and advances in data science, obtaining reliable models of such networks remains a very challenging problem. The scale of these emerging complex systems also poses a great difficulty. These obstacles also form a bottleneck for analyzing and engineering the dynamic structures (e.g., synchrony and clustering) and for controlling the collective behavior in such complex networks. This project will develop a unified data-driven framework to investigate fundamental questions regarding how to extract dynamics of a large-scale complex system or network from its simulation or measurement data, and how to control this system if the dynamics reconstruction is successful and reliable. The project will also support new initiatives to promote interdisciplinary education for students from traditionally underserved populations in local high schools in the city of St. Louis, MO, through the creation of summer research opportunities.By bridging systems and control theory with concepts and methods from algebraic geometry, time-series analysis, and machine learning, a unified data-driven framework will be established. Specifically, a novel approach based on spectral decomposition will be developed to extract the dynamics of a complex system and decode the topology of a complex network using its time-series data. The properties of the reconstructed network, e.g., connectivity and the coupling strength of nodes, will then be utilized to synthesize a dynamically-proximate reduced network that is tractable for control-theoretic analysis and design. Furthermore, novel topological and geometrical approaches will be derived to construct local and global embedding of high-dimensional data to low-dimensional manifolds, which will reveal hidden topological structures in large data sets and characterize transitions of flow of the underlying dynamical system. In collaboration with researchers in biology and chemistry, the network inference, dimensionality reduction, and control techniques will be applied to a diverse set of complex systems from cells to societies, for example, for decoding functional connectivity in cellular networks and analyzing social synchronization in groups of animals.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.