Machine Learning and Computational Mathematics

TL;DR

This paper discusses the impact of neural network-based machine learning on scientific computing and how computational mathematics can influence machine learning, aiming to foster integration between these fields.

Contribution

It provides a perspective on the mutual influence between machine learning and computational mathematics, highlighting recent progress and future directions.

Findings

Machine learning enables high-dimensional function approximation with high efficiency.

Impact of machine learning on scientific computing is significant and growing.

Computational mathematics can provide fundamental principles to improve machine learning.

Abstract

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.