CAREER: Optimal Transport and Dynamics in Machine Learning

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。机器学习的目标是开发能够在数据中找到有意义模式的算法。最初，这些算法为我们的数字生活带来了突破，从自动语言翻译到改进的在线搜索。它们越来越多地影响到生活的方方面面，从医学图像分析到欺诈检测。机器学习在科学领域也变得至关重要，因为研究人员使用相同的算法来分析数据集、测试假设和做出预测，从而加快了科学发现的步伐。然而，尽管取得了成功，机器学习的许多基本问题仍然没有得到很好的理解：神经网络惊人成功的背后是什么？它们什么时候可能会失败？如何使算法适合于单独的科学实验，利用几个世纪以来的特定领域知识来扩展分析的范围？为了回答这些问题，研究者将研究机器学习的数学基础，使用最优传输和偏微分方程的工具。这项研究将与本科生和研究生的教育机会相结合。研究者将举办本科生研究专题讨论会，以提高对校园研究机会的认识，目标是增加从事应用数学研究项目的不同学生的数量。研究人员还将开发一门关于最佳运输和机器学习的新研究生课程，该课程的讲座将向公众开放，并组织一个面向美国西部研究生的早期职业研究人员研讨会，这将为应用数学专业的学生提供一个向各种知名研究人员学习的机会，并展示他们自己的工作。这个研究计划的核心是三个主要项目。在第一个项目中，研究者将研究非局部相互作用在两层神经网络训练动力学中的作用，分析模型选择、数据分布和规则之间的相互作用如何影响鲁棒性和收敛到最优的速度。在第二个项目中，研究者将开发基于非线性扩散的采样和控制理论的粒子方法。在第三个项目中，研究者将使用新的最佳传输指标来开发可解释的机器学习方法，用于分析科学实验的多个组成部分的数据。这些方法随后将应用于粒子物理学中的机器学习任务，包括大型强子对撞机事件的分类。该奖项反映了美国国家科学基金会的法定使命，并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，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). The goal of machine learning is to develop algorithms that find meaningful patterns in data. Initially, such algorithms led to breakthroughs in our digital lives, from automated language translation to improved online search. Increasingly, they impact every aspect of life, from medical image analysis to fraud detection. Machine learning has also risen to paramount importance in the sciences, as researchers use the same algorithms to analyze datasets, test hypotheses, and make predictions, accelerating the pace of scientific discovery. However, despite its success, many foundational questions of machine learning remain poorly understood: What is behind the surprising success of neural networks and when might they fail? How can algorithms be tailored to individual scientific experiments, to leverage centuries of domain specific knowledge as they extend the reach of an analysis? To answer these questions, the investigator will study the mathematical foundations of machine learning, using tools from optimal transport and partial differential equations. This research will be integrated with educational opportunities for both undergraduate and graduate students. The investigator will hold undergraduate research symposia to improve awareness of campus research opportunities, with the goal of increasing the number of diverse students conducting research projects in applied mathematics. The investigator will also develop a new graduate course on optimal transport and machine learning, the lectures from which will be made publicly available, and organize an early-career researcher workshop geared to graduate students in the western United States, which will provide students in applied mathematics with an opportunity to learn from a diverse cadre of well-established researchers, as well as to present their own work.At the heart of this research program are three main projects. In the first project, the investigator will study the role of nonlocal interactions in the training dynamics of two-layer neural networks, analyzing how the interplay between model selection, data distribution, and regularity affects robustness and rate of convergence to optimum. In the second project, the investigator will develop particle methods for sampling and control theory based on nonlinear diffusions. In the third project, the investigator will use new optimal transport metrics to develop interpretable machine learning methods for analyzing data from multiple components of a scientific experiment. These methods will then be applied to machine learning tasks in particle physics, including classification of events at the Large Hadron Collider.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.