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.

翻译：不稳定性的存在（例如对抗样本）催生了一个高度活跃的研究领域，该领域致力于理解并增强神经网络的稳定性。本文聚焦于该领域中一个广受关注的分支，该分支借鉴了与连续动力系统及最优控制的联系，旨在为这一领域提供宏观视角。我们识别并阐述了支撑该领域现有工作的基本概念。随后，我们深入探讨了一种设计稳定神经网络的具体方法，发展了其理论背景，并描述了这些网络的实现方式。我们提供了实现该方法的代码，读者可对其进行调整和扩展。代码中还包含一个完整的图像分类对抗鲁棒性玩具示例笔记本，该示例无需读者计算机具备过高配置即可运行。最后，我们讨论了这一玩具示例，以便读者能在计算机上交互式地跟随操作。本工作将作为一本关于科学机器学习的书籍中的一章，该书目前正在修订中，主要面向学生读者。