  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.
