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

职业：通过利用低维结构解决网络交互动力系统的估计问题：数学基础、算法和应用

• 批准号: 2340631
• 负责人: Sui Tang
• 金额: $ 44.94万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2029-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2340631&HistoricalAwards=false
• 关键词: CAREER; Solving; Estimation; Problems; Networked

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.

网络交互动力系统 (NetID) 无处不在，显示出由代理或粒子相互作用产生的复杂行为，这些系统已在生态学、工程和社会科学等多个领域得到应用，但其高维性质使其具有挑战性。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 模式可能源自低维交互。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。总体目标是创建有效的、物理上可解释的替代模型，以弥合定性分析和分析之间的差距。该研究项目将为定量本科生和研究生提供研究机会，其中包括 NetID 和机器学习交叉领域的研究生暑期学校。将特别注重参与该项目的研究结果还将丰富本科生和研究生教育的数学数据科学课程材料。该项目旨在推动数学基础、统计和计算方面的进步，以解决这一问题。 NetID 的估计问题将集中在三个主要领域：（1）通过利用其固有的稀疏性、平滑性和低秩性等低维性，开发创新的采样策略，以实现具有线性交互的 NetID 的最佳数据恢复。 (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交互作用的 NetID 的稳健统计估计，创建物理一致的估计器，绕过“维数灾难”，同时探索样本大小的可识别性和收敛性(3) 研究图神经微分方程的统计预测特性，旨在推导其可传递性和泛化误差的上限。该项目的结果有望解决以下问题的计算挑战。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。