CAREER: Sparse Model Selection for Nonlinear Evolution Equations

从静态和/或动态数据中提取信息是许多科学和工业问题中的一项重要任务；包括但不限于机器学习、数据挖掘、图像处理和科学数据的自动分析。这个项目的重点是学习生成观测数据的基本过程，从某种意义上说，这是从数据中“逆向工程”模型。这些模型经常被用来获得对数据的洞察(例如，从实验观察中确定数学原理)或做出基于数据的决策(例如，趋势预测)。这是一个具有挑战性的数学和计算问题，因为人们通常事先关于该过程的信息有限，而实际数据通常是噪声和/或不完整的。研究目标是构造学习母函数的有效计算方法。这将涉及以优化和抽样理论为中心的各种数学技术。教育目标是为本科生和研究生提供高级培训，为他们进入美国STEM劳动力市场做好准备。特别是，学生将通过数学和计算研究项目、协作暑期计划、工作组和结合教育和研究的高级课程来指导和培训学生。目标是开发用于模型学习、数据分析和其他机器学习任务的计算方法。总体目标包括：(I)使用稀疏性、平稳性和随机性来补充学习的优化模型的构建，(Ii)高效且可证明收敛的数值方法的设计，(Iii)对样本大小和离群值稳健的方法的发展，以及(Iv)为本科生和研究生创建和实施融合教育和研究的活动。该奖项反映了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。

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.