Machine Learning and Computational Mathematics (Communications in Computational Physics, 2020)
Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This hasopened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the sametime, machine learning has also acquired the reputation of being a set of "black box"type of tricks, without fundamental principles. This has been a real obstacle for makingfurther progress in machine learning.In this article, we try to address the following two very important questions: (1)How machine learning has already impacted and will further impact computationalmathematics, scientific computing and computational science? (2) How computationalmathematics, particularly numerical analysis, can impact machine learning? We describe some of the most important progress that has been made on these issues. Ourhope is to put things into a perspective that will help to integrate machine learningwith computational mathematics.