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，表现出复杂的行为，从代理或粒子的相互作用。这些系统已经在不同的领域，包括生态学，工程学和社会科学中找到了应用，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，称为“维数灾难”。应用数学的最新进展揭示了这些复杂性，揭示了复杂的NetID模式可以从低维交互中产生。基于这些见解，该项目致力于开发一个理论和计算框架，通过利用底层低维结构来解决这些模型中的估计问题。总体目标是创建有效的，物理上可解释的代理模型，弥合定性分析和定量数据驱动的应用程序之间的差距，从传感器网络优化到模拟环境和气候对鱼类迁徙的影响。该研究计划将为本科生和研究生提供研究机会，在NetID和机器学习的交叉点上开设研究生暑期学校。将特别关注女性和代表性不足的少数民族学生参与这个充满活力的领域，将机器学习与微分方程相结合。该项目的研究成果还将丰富本科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的估计问题提供基础数学，统计和计算方面的进步。研究将集中在三个主要领域：（1）开发创新的采样策略，优化数据恢复的NetID与线性交互，利用其固有的低维稀疏性，平滑性，低秩。(2)通过结合机器学习、数值分析和功能数据分析来建立具有非线性时变交互的NetID的稳健统计估计，以创建绕过“维数灾难”的物理一致的估计器，同时探索随着样本量增加的可识别性和收敛性。(3)研究了图神经微分方程的统计预测性质，旨在推导其可传递性和推广误差的上界。该项目的成果有望解决大规模图神经网络的计算挑战，并在NetID研究中搭建理论和实践的桥梁。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。

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.