Collaborative Research: Friedrichs Learning: Mathematical Foundation and Applications

通过深度学习的人工智能（AI）已经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。该项目旨在为应用于生物学和工程学的新型人工智能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，将机器学习、数学分析和计算科学中的几个学科联系起来。该项目还将提供培训机会，加强未来的劳动力队伍。该项目旨在开发一个数学框架，用于弱意义上的机器学习问题的极大极小优化。这个学习问题设置的灵感来自于Friedrichs的偏微分方程系统理论的开创性工作，这被称为Friedrichs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，极大极小设置下深度神经网络的最优收敛，以及弱形式下深度学习的泛化分析。结果将提供新的见解来解释测试函数在弱形式中的作用。研究结果将进一步推动深度学习技术的发展，以更可靠的预测和计算，以原则性的方式进行更好的风险评估，而不是反复试验。本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。

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.