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

网络相互作用动力系统（netid）无处不在，显示由代理或粒子相互作用产生的复杂行为。这些系统已经在不同的领域得到了应用，包括生态学、工程学和社会科学，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，被称为“维度的诅咒”。应用数学的最新进展揭示了这些复杂性，揭示了复杂的NetID模式可以从低维相互作用中产生。在这些见解的基础上，该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。总体目标是创建高效、物理可解释的替代模型，弥合定性分析和定量数据驱动应用之间的差距，从传感器网络优化到模拟环境和气候对鱼类迁徙的影响。该研究项目将为本科生和研究生提供研究机会，并在netid和机器学习的交叉点开设研究生暑期学校。将特别关注让女性和代表性不足的少数族裔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。该项目的研究结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的估计问题提供基本的数学、统计和计算方面的进展。研究将集中在三个主要领域：(1)通过利用具有线性相互作用的netid在稀疏性、平滑性和低秩性方面固有的低维性，开发创新的采样策略，以实现数据的最佳恢复。(2)通过结合机器学习、数值分析和功能数据分析，建立具有非线性时变相互作用的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”，同时探索随样本量增加的可识别性和收敛性。(3)研究图神经微分方程的统计预测性质，旨在推导其可转移性和泛化误差的上界。该项目的结果有望解决大规模图神经网络的计算挑战，并在netid研究中架起理论与实践的桥梁。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。

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.