# Stable neural networks and connections to continuous dynamical systems

arXiv: https://arxiv.org/abs/2510.22299 · HTML: https://mlsanity.com/paper/2510.22299/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 37 references that are in the map.
- First posted on arXiv 25 Oct 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [A Dynamical System Perspective for Lipschitz Neural Networks](https://mlsanity.com/paper/2110.12690/) (2021)
- [Convolutional Proximal Neural Networks and Plug-and-Play Algorithms](https://mlsanity.com/paper/2011.02281/) (2020)
- [CLIP: Cheap Lipschitz Training of Neural Networks](https://mlsanity.com/paper/2103.12531/) (2021)
- [Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Design](https://mlsanity.com/paper/2105.13205/) (2021)
- [1-Lipschitz Layers Compared: Memory, Speed, and Certifiable Robustness](https://mlsanity.com/paper/2311.16833/) (2023)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Dynamical systems' based neural networks](https://mlsanity.com/paper/2210.02373/) (2022)
- [Designing Stable Neural Networks using Convex Analysis and ODEs](https://mlsanity.com/paper/2306.17332/) (2023)
- [FI-ODE: Certifiably Robust Forward Invariance in Neural ODEs](https://mlsanity.com/paper/2210.16940/) (2022)

## Abstract (arXiv)

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.