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Collaborative Research: Friedrichs Learning: Mathematical Foundation and Applications

合作研究：弗里德里希学习：数学基础与应用

基本信息

批准号：
2206333
负责人：
Haizhao Yang
金额：
$ 12.59万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206333&HistoricalAwards=false
关键词：
Collaborative Research Friedrichs Learning Mathematical

项目摘要

Artificial intelligence (AI) via deep learning has revolutionized the development of science and engineering, addressing many challenges in biology, material sciences, chemistry, and other areas. Despite its importance, the mathematical foundation of AI is still underdeveloped. The project aims to develop a mathematical foundation for novel AI learning algorithms in applications to biology and engineering. Novel theories and learning algorithms will advance computational and data-based methods, connecting several disciplines in machine learning, mathematical analysis, and computational science. The project will also provide training opportunities and strengthen future workforces. The project aims to develop a mathematical framework for a minimax optimization of a machine learning problem in the weak sense. This learning problem setting is inspired by the seminal work of Friedrichs’ theory for partial differential equation systems, which is called Friedrichs learning. The research will focus on gradient-based methods to solve the minimax optimization problem. The investigators will establish a systematic theoretical framework, including approximation theory in the weak form, optimal convergence for deep neural networks in the minimax setting, and generalization analysis of deep learning in the weak form. The results will provide new insights to explain the role of test functions in the weak form. The results will further advance deep learning techniques with more reliable predictions and computation with a better risk assessment in a principled manner instead of trial and error. The theoretical analysis in this project will be the building block of new theories in the strong form of standard deep learning techniques in the literature.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.

通过深度学习实现的人工智能（AI）彻底改变了科学和工程的发展，解决了生物学、材料科学、化学和其他领域的许多挑战。尽管人工智能很重要，但它的数学基础仍然不发达。该项目旨在为生物学和工程应用中的新型AI学习算法开发数学基础。新的理论和学习算法将推进计算和基于数据的方法，连接机器学习，数学分析和计算科学中的几个学科。该项目还将提供培训机会，加强未来的劳动力队伍。该项目旨在为弱意义上的机器学习问题的极大极小优化开发一个数学框架。这个学习问题设置的灵感来自于弗里德里希的偏微分方程系统理论的开创性工作，这被称为弗里德里希学习。研究将集中在基于梯度的方法来解决极大极小优化问题。研究人员将建立一个系统的理论框架，包括弱形式的近似理论，minimax设置下深度神经网络的最优收敛，以及弱形式的深度学习的泛化分析。结果将提供新的见解来解释弱形式的测试函数的作用。这些结果将进一步推进深度学习技术，使其具有更可靠的预测和计算，并以原则性的方式进行更好的风险评估，而不是反复试验。该项目中的理论分析将成为文献中标准深度学习技术强有力形式的新理论的基石。该奖项反映了NSF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而被认为值得支持。