CAREER: Solving Estimation Problems of Networked Interacting Dynamical Systems Via Exploiting Low Dimensional Structures: Mathematical Foundations, Algorithms and Applications

网络交互动力系统(NetID)是无处不在的，显示了由代理或粒子的相互作用引起的复杂行为。这些系统已经在包括生态学、工程学和社会科学在内的不同领域得到了应用，但它们的高维性质使它们的研究具有挑战性。这通常会导致重大的理论和计算困难，即所谓的“维度诅咒”。应用数学的最新进展揭示了这些复杂性，揭示了复杂的NetID模式可以产生于低维相互作用。在这些洞察的基础上，这个项目致力于开发一个理论和计算框架，通过利用潜在的低维结构来解决这些模型中的估计问题。首要目标是创建有效的、物理上可解释的替代模型，以弥合定性分析和定量数据驱动应用之间的差距，范围从传感器网络优化到对鱼类迁徙的环境和气候影响建模。该研究计划将为本科生和研究生提供研究机会，特色是在NetID和机器学习的交叉点开设研究生暑期班。将特别关注在这个充满活力的领域吸引女性和未被充分代表的少数族裔学生，将机器学习与微分方程式结合起来。该项目的发现还将丰富本科生和研究生教育的数学数据科学课程材料。该项目旨在为解决NETID的估计问题取得基础的数学、统计和计算方面的进展。研究将集中在三个主要方面：(1)通过利用线性交互网络入侵检测系统固有的稀疏性、光滑性、低秩性等特性，开发创新的采样策略，以实现线性交互网络入侵检测系统的最优数据恢复。(2)通过将机器学习、数值分析和函数数据分析相结合，建立具有非线性时变交互作用的NetID的稳健统计估计，以创建物理上一致的估计器，绕过“维度诅咒”，同时探索随着样本量的增加可辨识性和收敛。(3)研究图神经微分方程解的统计预测性质，得到其可转移性和泛化误差的上界。该项目的成果有望解决大规模图形神经网络的计算挑战，并在NetIDs研究中架起理论和实践的桥梁。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。

Networked Interacting Dynamical Systems (NetIDs) are ubiquitous, displaying complex behaviors that arise from the interactions of agents or particles. These systems have found applications in diverse fields, including ecology, engineering, and social sciences, yet their high-dimensional nature makes them challenging to study. This often leads to significant theoretical and computational difficulties, known as the “curse of dimensionality.” Recent advances in applied mathematics have shed light on these complexities, revealing that complex NetID patterns can arise from low dimensional interactions. Building on these insights, this project is dedicated to developing a theoretical and computational framework to address the estimation problems within these models by exploiting the underlying low dimensional structures. The overarching goal is to create efficient, physically interpretable surrogate models that bridge the gap between qualitative analysis and quantitative data-driven applications, ranging from sensor network optimization to modeling the environmental and climate impacts on fish migration. This research program will provide research opportunities for both undergraduate and graduate students, featuring a graduate summer school at the intersection of NetIDs and machine learning. There will be a particular focus on engaging female and underrepresented minority students in this vibrant field, blending machine learning with differential equations. The project's findings will also enrich mathematical data science course materials for both undergraduate and graduate education.This project aims to make fundamental mathematical, statistical, and computational advances for solving NetIDs' estimation problems. The research will focus on three primary areas: (1) Developing innovative sampling strategies for optimal data recovery in NetIDs with linear interactions by exploiting their inherent low-dimensionality in terms of sparsity, smoothness, low-rankness. (2) Establishing robust statistical estimation of NetIDs with nonlinear time-varying interactions by combining machine learning, numerical analysis, and functional data analysis to create physically consistent estimators that bypass the “curse of dimensionality,” while exploring the identifiability and convergence as sample sizes increase. (3) Investigating the statistical predictive properties of Graph Neural Differential Equations, aiming to derive upper bounds for their transferability and generalization error. The results of this project are expected to address the computational challenges of large-scale Graph Neural Networks and bridge theory and practice in NetIDs research.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.