Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't

TL;DR

This paper reviews recent mathematical insights into neural network-based machine learning, highlighting key achievements, open problems, and the importance of combining rigorous results with numerical experiments to deepen understanding.

Contribution

It provides a comprehensive overview of mathematical results, experimental insights, and simplified models, along with open problems, to advance understanding of neural networks.

Findings

Summarizes key mathematical results in neural network theory

Highlights the role of numerical experiments in understanding neural networks

Identifies important open problems for future research

Abstract

The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied mathematics, we will not only give attention to rigorous mathematical results, but also the insight we have gained from careful numerical experiments as well as the analysis of simplified models. Along the way, we also list the open problems which we believe to be the most important topics for further study. This is not a complete overview over this quickly moving field, but we hope to provide a perspective which may be helpful especially to new researchers in the area.