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

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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.