# Robust Design of Deep Neural Networks against Adversarial Attacks based on Lyapunov Theory

arXiv: https://arxiv.org/abs/1911.04636 · HTML: https://mlsanity.com/paper/1911.04636/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Nov 2019.

## Built on by (most cited first, 13 of 13)

- [Advances in adversarial attacks and defenses in computer vision: A survey](https://mlsanity.com/paper/2108.00401/) (2021)
- [Learning Diverse-Structured Networks for Adversarial Robustness](https://mlsanity.com/paper/2102.01886/) (2021)
- [Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity](https://mlsanity.com/paper/2111.12906/) (2021)
- [Adversarially Robust Out-of-Distribution Detection Using Lyapunov-Stabilized Embeddings](https://mlsanity.com/paper/2410.10744/) (2024)
- [The Uncanny Valley: Exploring Adversarial Robustness from a Flatness Perspective](https://mlsanity.com/paper/2405.16918/) (2024)
- [Jacobian Norm with Selective Input Gradient Regularization for Improved and Interpretable Adversarial Defense](https://mlsanity.com/paper/2207.13036/) (2022)
- [Topological derivative approach for deep neural network architecture adaptation](https://mlsanity.com/paper/2502.06885/) (2025)
- [Stop Walking in Circles! Bailing Out Early in Projected Gradient Descent](https://mlsanity.com/paper/2503.19347/) (2025)
- [Adversarial Machine Learning Attacks on Financial Reporting via Maximum Violated Multi-Objective Attack](https://mlsanity.com/paper/2507.05441/) (2025)
- [Understanding and Improving Adversarial Robustness of Neural Probabilistic Circuits](https://mlsanity.com/paper/2509.20549/) (2025)
- [Random Spiking Neural Networks are Stable and Spectrally Simple](https://mlsanity.com/paper/2511.00904/) (2025)
- [Ortho-ODE: Enhancing Robustness and of Neural ODEs against Adversarial Attacks](https://mlsanity.com/paper/2305.09179/) (2023)
- [Adversarial Samples Are Not Created Equal](https://mlsanity.com/paper/2601.00577/) (2026)

## Abstract (arXiv)

Deep neural networks (DNNs) are vulnerable to subtle adversarial perturbations applied to the input. These adversarial perturbations, though imperceptible, can easily mislead the DNN. In this work, we take a control theoretic approach to the problem of robustness in DNNs. We treat each individual layer of the DNN as a nonlinear dynamical system and use Lyapunov theory to prove stability and robustness locally. We then proceed to prove stability and robustness globally for the entire DNN. We develop empirically tight bounds on the response of the output layer, or any hidden layer, to adversarial perturbations added to the input, or the input of hidden layers. Recent works have proposed spectral norm regularization as a solution for improving robustness against l2 adversarial attacks. Our results give new insights into how spectral norm regularization can mitigate the adversarial effects. Finally, we evaluate the power of our approach on a variety of data sets and network architectures and against some of the well-known adversarial attacks.