Neural network-based deep 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, deep 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 talk, I will try to address the following two questions:

1. How machine learning will impact computational mathematics and computational science?
2. How computational mathematics, particularly numerical analysis, can impact machine learning? We describe some of the most important progresses that have been made on these issues so far. Our hope is to put things into a perspective that will help to integrate machine learning with computational science.