CAREER: Exploiting Low-Dimensional Structures in Data Science: Manifold Learning, Partial Differential Equation Identification, and Neural Networks

职业：在数据科学中利用低维结构：流形学习、偏微分方程识别和神经网络

• 批准号: 2145167
• 负责人: Wenjing Liao
• 金额: $ 48.14万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-03-01 至 2027-02-28
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2145167&HistoricalAwards=false
• 关键词: CAREER; Exploiting; Low; Dimensional; Structures

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). In general, scientific and engineering data can be high-dimensional, but in many practical applications, data exhibit low-dimensional features due to local regularities, global symmetries, or repetitive patterns. This project aims to develop new theoretical and computational tools to exploit low-dimensional structures in data science. The overall goal is to develop improved computational algorithms for machine learning with high-dimensional datasets that have additional structure. Machine learning research will also be integrated with data science education, including a bridge program that aims to help prepare undergraduate students with diverse backgrounds for careers in both industry and academia.This project aims to make fundamental mathematical, statistical, and computational advances in analysis of high-dimensional data with structures. Research directions include manifold learning, identification of partial differential equations, and a nonparametric estimation theory for neural networks. This work focuses on three sets of related but distinct questions. The first set is about efficient approximation of functions supported on and near a low-dimensional manifold. Efficient algorithms will be developed to build local linear approximations of the manifold and polynomial approximations of the function. A theoretical goal is to prove that the function estimation error converges to zero as the sample size grows with a fast rate depending on the intrinsic dimension of the manifold. The second set is on robust PDE identification from noisy data. The PI will combine tools in machine learning and numerical PDEs to explore noisy data and robustly identify the underlying PDE and dynamics. This project will address denoising, recovery of spatially varying parameters, and kernel identification in nonlocal equations. The third set of questions concerns nonparametric estimation theory for neural networks for learning operators between infinite dimensional function spaces. The work aims to provide an upper bound for the error in estimation of Lipschitz operators.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.

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。一般来说，科学和工程数据可以是高维的，但在许多实际应用中，数据表现出低维特征，由于局部对称性，全局对称性或重复模式。该项目旨在开发新的理论和计算工具，以利用数据科学中的低维结构。总体目标是开发改进的计算算法，用于具有额外结构的高维数据集的机器学习。机器学习研究还将与数据科学教育相结合，包括一个桥梁项目，旨在帮助具有不同背景的本科生为工业和学术界的职业生涯做好准备。该项目旨在在分析具有结构的高维数据方面取得基础数学，统计和计算方面的进步。研究方向包括流形学习，偏微分方程的识别和神经网络的非参数估计理论。这项工作的重点是三组相关但不同的问题。第一个集合是关于低维流形上和附近支持的函数的有效逼近。将开发有效的算法来建立流形的局部线性近似和函数的多项式近似。一个理论目标是证明函数估计误差收敛到零的样本大小增长的速度取决于流形的内在维度的快速。第二组是从噪声数据中进行鲁棒PDE识别。PI将结合机器学习和数值PDE中的联合收割机工具，以探索噪声数据并鲁棒地识别潜在的PDE和动态。这个项目将解决去噪，恢复空间变化的参数，并在非局部方程核识别。第三组问题涉及神经网络学习无限维函数空间之间的算子的非参数估计理论。这项工作旨在为Lipschitz算子的估计误差提供一个上限。该奖项反映了NSF的法定使命，并被认为值得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来支持。

项目成果

WeakIdent: Weak formulation for identifying differential equation using narrow-fit and trimming

WeakIdent：使用窄拟合和修剪识别微分方程的弱公式

• DOI: 10.1016/j.jcp.2023.112069
• 发表时间: 2023
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Tang, Mengyi;Liao, Wenjing;Kuske, Rachel;Kang, Sung Ha
• 通讯作者: Kang, Sung Ha

Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT)

变系数微分方程辨识的群投影子空间追踪 (GP-IDENT)

• DOI: 10.1016/j.jcp.2023.112526
• 发表时间: 2023
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: He, Yuchen;Kang, Sung Ha;Liao, Wenjing;Liu, Hao;Liu, Yingjie
• 通讯作者: Liu, Yingjie

Robust Identification of Differential Equations by Numerical Techniques from a Single Set of Noisy Observation

• DOI: 10.1137/20m134513x
• 发表时间: 2020-06
• 期刊: SIAM J. Sci. Comput.
• 作者: Yuchen He;S. Kang;Wenjing Liao;Hao Liu;Yingjie Liu

Multiscale regression on unknown manifolds

• DOI: 10.3934/mine.2022028
• 发表时间: 2021-01
• 期刊: ArXiv
• 作者: Wenjing Liao;M. Maggioni;S. Vigogna

Deep nonparametric estimation of intrinsic data structures by chart autoencoders: Generalization error and robustness

通过图表自动编码器对内在数据结构进行深度非参数估计：泛化误差和鲁棒性

• DOI: 10.1016/j.acha.2023.101602
• 发表时间: 2024
• 期刊: Applied and Computational Harmonic Analysis
• 影响因子: 2.5
• 作者: Liu, Hao;Havrilla, Alex;Lai, Rongjie;Liao, Wenjing
• 通讯作者: Liao, Wenjing