# Tracking Finite-Time Lyapunov Exponents to Robustify Neural ODEs

arXiv: https://arxiv.org/abs/2602.09613 · HTML: https://mlsanity.com/paper/2602.09613/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 10 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Towards Robust Neural Networks via Random Self-ensemble](https://mlsanity.com/paper/1712.00673/) (2017)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [A Unified Gradient Regularization Family for Adversarial Examples](https://mlsanity.com/paper/1511.06385/) (2015)
- [Lyapunov spectra of chaotic recurrent neural networks](https://mlsanity.com/paper/2006.02427/) (2020)
- [On Lyapunov Exponents for RNNs: Understanding Information Propagation Using Dynamical Systems Tools](https://mlsanity.com/paper/2006.14123/) (2020)
- [Finite-time Lyapunov exponents of deep neural networks](https://mlsanity.com/paper/2306.12548/) (2023)
- [On the regularized risk of distributionally robust learning over deep neural networks](https://mlsanity.com/paper/2109.06294/) (2021)
- [Learning across scales - A multiscale method for Convolution Neural Networks](https://mlsanity.com/paper/1703.02009/) (2017)
- [Gradient Flossing: Improving Gradient Descent through Dynamic Control of Jacobians](https://mlsanity.com/paper/2312.17306/) (2023)
- [On Lyapunov exponents and adversarial perturbation](https://mlsanity.com/paper/1802.06927/) (2018)

## Built on by (most cited first, 1 of 1)

- [Dynamics to decision: A mathematical theory of Lyapunov spectra and decision boundaries in deep classifiers](https://mlsanity.com/paper/2609.39190/) (2026)

## Abstract (arXiv)

We investigate finite-time Lyapunov exponents (FTLEs), a measure for exponential separation of input perturbations, of deep neural networks within the framework of continuous-depth neural ODEs. We demonstrate that FTLEs are powerful organizers for input-output dynamics, allowing for better interpretability and the comparison of distinct model architectures. We establish a direct connection between Lyapunov exponents and adversarial vulnerability, and propose a novel training algorithm that improves robustness by FTLE regularization. The key idea is to suppress exponents far from zero in the early stage of the input dynamics. This approach enhances robustness and reduces computational cost compared to full-interval regularization, as it avoids a full ``double'' backpropagation.