Machine Learning and Computational Mathematics
It tackles the integration of machine learning with computational mathematics to overcome the 'black box' reputation and advance both fields, but it is incremental as it reviews existing progress.
The paper addresses the impact of machine learning on computational mathematics and vice versa, aiming to integrate these fields by describing progress on their mutual influence.
Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, 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 making further 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 computational mathematics, scientific computing and computational science? (2) How computational mathematics, particularly numerical analysis, {can} impact machine learning? We describe some of the most important progress that has been made on these issues. Our hope is to put things into a perspective that will help to integrate machine learning with computational mathematics.