CAREER: Sparse Model Selection for Nonlinear Evolution Equations

Extracting information from stationary and/or dynamic data is an important task in many scientific and industrial problems; including but not limited to, machine learning, data mining, image processing, and automated analysis of scientific data. This project focuses on learning the underlying process that generates observational data, in a sense, "reverse-engineering" models from data. These models are often used to gain insights on the data (for example, determining mathematical principles from experimental observations) or to make data-enabled decisions (for example, trend prediction). This is a challenging mathematical and computational problem, since one often has limited information on the process beforehand and real data is often noisy and/or incomplete. The research objective is to construct efficient computational methods for learning generating functions. This will involve a variety of mathematical techniques centered around optimization and sampling theory. The educational objective is to provide advanced training to undergraduate and graduate students in order to prepare them for the U.S. STEM workforce. In particular, students will be mentored and trained through mathematical and computational research projects, collaborative summer programs, working groups, and advanced courses that integrate education and research. The goal is to develop computational methods for model learning, data analysis, and other machine learning tasks. The overall objectives include: (i) the construction of optimization models that use sparsity, smoothness, and randomness to supplement the learning, (ii) the design of efficient and provably convergent numerical methods, (iii) the development of methods that are robust to sample size and outliers, and (iv) the creation and implementation of activities for undergraduate and graduate students that integrate education and 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.