Stable neural networks and connections to continuous dynamical systems

TL;DR

This paper reviews the connection between stable neural networks and continuous dynamical systems, providing theoretical insights, implementation code, and a toy example on adversarial robustness for educational purposes.

Contribution

It offers a comprehensive overview of stability concepts in neural networks linked to dynamical systems, and presents a specific approach with implementation details and an interactive toy example.

Findings

Theoretical foundation for stable neural networks connected to dynamical systems.

Implementation code and interactive notebook for adversarial robustness.

Educational resource included in a book chapter on scientific machine learning.

Abstract

The existence of instabilities, for example in the form of adversarial examples, has given rise to a highly active area of research concerning itself with understanding and enhancing the stability of neural networks. We focus on a popular branch within this area which draws on connections to continuous dynamical systems and optimal control, giving a bird's eye view of this area. We identify and describe the fundamental concepts that underlie much of the existing work in this area. Following this, we go into more detail on a specific approach to designing stable neural networks, developing the theoretical background and giving a description of how these networks can be implemented. We provide code that implements the approach that can be adapted and extended by the reader. The code further includes a notebook with a fleshed-out toy example on adversarial robustness of image classification that can be run without heavy requirements on the reader's computer. We finish by discussing this toy example so that the reader can interactively follow along on their computer. This work will be included as a chapter of a book on scientific machine learning, which is currently under revision and aimed at students.