# Robust Classification using Contractive Hamiltonian Neural ODEs

arXiv: https://arxiv.org/abs/2203.11805 · HTML: https://mlsanity.com/paper/2203.11805/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 28 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## 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)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Adversarial Attacks and Defenses in Images, Graphs and Text: A Review](https://mlsanity.com/paper/1909.08072/) (2019)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [Towards the first adversarially robust neural network model on MNIST](https://mlsanity.com/paper/1805.09190/) (2018)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [On Robustness of Neural Ordinary Differential Equations](https://mlsanity.com/paper/1910.05513/) (2019)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Stable Neural Flows](https://mlsanity.com/paper/2003.08063/) (2020)
- [Robust Implicit Networks via Non-Euclidean Contractions](https://mlsanity.com/paper/2106.03194/) (2021)
- [Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Design](https://mlsanity.com/paper/2105.13205/) (2021)
- [Contracting Implicit Recurrent Neural Networks: Stable Models with Improved Trainability](https://mlsanity.com/paper/1912.10402/) (2019)
- [An Ode to an ODE](https://mlsanity.com/paper/2006.11421/) (2020)
- [Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity](https://mlsanity.com/paper/2111.12906/) (2021)

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

- [Dynamical systems' based neural networks](https://mlsanity.com/paper/2210.02373/) (2022)
- [Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations](https://mlsanity.com/paper/2304.02976/) (2023)
- [Learning Robust State Observers using Neural ODEs (longer version)](https://mlsanity.com/paper/2212.00866/) (2022)
- [Physically Consistent Neural ODEs for Learning Multi-Physics Systems](https://mlsanity.com/paper/2211.06130/) (2022)
- [Efficient Certified Training and Robustness Verification of Neural ODEs](https://mlsanity.com/paper/2303.05246/) (2023)
- [Neural-Rendezvous: Provably Robust Guidance and Control to Encounter Interstellar Objects](https://mlsanity.com/paper/2208.04883/) (2022)
- [Universal Approximation Property of Hamiltonian Deep Neural Networks](https://mlsanity.com/paper/2303.12147/) (2023)
- [Resilient Graph Neural Networks: A Coupled Dynamical Systems Approach](https://mlsanity.com/paper/2311.06942/) (2023)
- [ICODE: Modeling Dynamical Systems with Extrinsic Input Information](https://mlsanity.com/paper/2411.13914/) (2024)
- [L2RU: a Structured State Space Model with prescribed L2-bound](https://mlsanity.com/paper/2503.23818/) (2025)
- [PAC bounds of continuous Linear Parameter-Varying systems related to neural ODEs](https://mlsanity.com/paper/2307.03630/) (2023)
- [Robust Convolution Neural ODEs via Contractivity-promoting regularization](https://mlsanity.com/paper/2508.11432/) (2025)
- [Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness](https://mlsanity.com/paper/2509.21879/) (2025)

## Abstract (arXiv)

Deep neural networks can be fragile and sensitive to small input perturbations that might cause a significant change in the output. In this paper, we employ contraction theory to improve the robustness of neural ODEs (NODEs). A dynamical system is contractive if all solutions with different initial conditions converge to each other exponentially fast. As a consequence, perturbations in initial conditions become less and less relevant over time. Since in NODEs the input data corresponds to the initial condition of dynamical systems, we show contractivity can mitigate the effect of input perturbations. More precisely, inspired by NODEs with Hamiltonian dynamics, we propose a class of contractive Hamiltonian NODEs (CH-NODEs). By properly tuning a scalar parameter, CH-NODEs ensure contractivity by design and can be trained using standard backpropagation. Moreover, CH-NODEs enjoy built-in guarantees of non-exploding gradients, which ensure a well-posed training process. Finally, we demonstrate the robustness of CH-NODEs on the MNIST image classification problem with noisy test data.