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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。通常，科学和工程数据可以是高维的，但在许多实际应用中，由于局部规则性、全局对称性或重复模式，数据表现出低维特征。该项目旨在开发新的理论和计算工具，以开发数据科学中的低维结构。总体目标是利用具有额外结构的高维数据集开发用于机器学习的改进计算算法。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行业和学术界的职业生涯做准备。该项目旨在使高维数据与结构分析的基础数学、统计和计算方面取得进展。研究方向包括流形学习、偏微分方程的辨识和神经网络的非参数估计理论。这项工作集中在三组相关但不同的问题上。第一组是关于低维流形上及其附近所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随着样本大小的快速增长而收敛到零，这取决于流形的固有维度。第二组是关于从噪声数据中进行稳健的偏微分方程识别。PI将结合机器学习和数值PDE中的工具来探索噪声数据，并强有力地识别潜在的PDE和动力学。这个项目将解决去噪，空间变化参数的恢复，以及非局部方程中的核识别。第三组问题涉及无穷维函数空间之间学习算子的神经网络的非参数估计理论。这项工作旨在为Lipschitz运算符的估计误差提供上限。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.