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.

网络相互作用的动态系统（NETID）无处不在，表现出由代理或颗粒的相互作用引起的复杂行为。这些系统在包括生态学，工程和社会科学在内的潜水领域中发现了应用，但是它们的高维质使它们挑战了学习。这通常会导致重大的理论和计算困难，称为“维度的诅咒”。应用数学的最​​新进展已经阐明了这些复杂性，揭示了复杂的Netid模式可以由低维相互作用引起。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，以通过利用基本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，可解释的替代模型，以弥合定性分析和定量数据驱动的应用之间的差距，从传感器网络优化到建模环境以及气候对鱼类迁移的影响。该研究计划将为本科生和研究生提供研究机会，并在Netids和机器学习的交叉点上为暑期学校提供。在这个充满活力的领域中，将尤其关注女性和代表性不足的少数族裔学生，将机器学习与微分方程融合在一起。项目的发现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Netids的估计问题提供基本的数学，统计和计算进步。该研究将重点关注三个主要领域：（1）通过利用其继承的低差异性在稀疏性，平稳性，低级别方面，开发具有线性相互作用的NetID的创新抽样策略，以实现线性相互作用的最佳数据恢复。 （2）通过结合机器学习，数值分析和功能数据分析来建立具有非线性时间变化相互作用的NetID的稳健统计估计，以创建绕过“维数的诅咒”的物理一致的估计器，同时探索识别和收敛作为样本量增加的识别和收敛性。 （3）研究图神经微分方程的统计预测性能，旨在为其可传递性和泛化误差提供上限。预计该项目的结果将解决NetIDS研究中大型图形神经网络以及桥梁理论和实践的计算挑战。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子和更广泛的影响评估标准来评估NSF的法定任务。