Collaborative Research: Friedrichs Learning: Mathematical Foundation and Applications

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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.