# Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2304.02976 · HTML: https://mlsanity.com/paper/2304.02976/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 35 references that are in the map.
- First posted on arXiv 6 Apr 2023.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Seq2Sick: Evaluating the Robustness of Sequence-to-Sequence Models with Adversarial Examples](https://mlsanity.com/paper/1803.01128/) (2018)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Stable Neural Flows](https://mlsanity.com/paper/2003.08063/) (2020)
- [Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Design](https://mlsanity.com/paper/2105.13205/) (2021)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Youla-REN: Learning Nonlinear Feedback Policies with Robust Stability Guarantees](https://mlsanity.com/paper/2112.01253/) (2021)
- [Contracting Nonlinear Observers: Convex Optimization and Learning from Data](https://mlsanity.com/paper/1711.08135/) (2017)

## Built on by (most cited first, 8 of 8)

- [Learning to Boost the Performance of Stable Nonlinear Systems](https://mlsanity.com/paper/2405.00871/) (2024)
- [Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recovery](https://mlsanity.com/paper/2412.07544/) (2024)
- [Learning Stable and Passive Neural Differential Equations](https://mlsanity.com/paper/2404.12554/) (2024)
- [Neural Port-Hamiltonian Models for Nonlinear Distributed Control: An Unconstrained Parametrization Approach](https://mlsanity.com/paper/2411.10096/) (2024)
- [RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustness](https://mlsanity.com/paper/2306.12612/) (2023)
- [On Dissipativity of Cross-Entropy Loss in Training ResNets](https://mlsanity.com/paper/2405.19013/) (2024)
- [Circuit realization and hardware linearization of monotone operator equilibrium networks](https://mlsanity.com/paper/2509.13793/) (2025)
- [LipSSM: Structurally Lipschitz-Bounded Cascaded State-Space Model via Metric Transfer between Consecutive SSM Layers](https://mlsanity.com/paper/2609.30973/) (2026)

## Abstract (arXiv)

In this work, we introduce and study a class of Deep Neural Networks (DNNs) in continuous-time. The proposed architecture stems from the combination of Neural Ordinary Differential Equations (Neural ODEs) with the model structure of recently introduced Recurrent Equilibrium Networks (RENs). We show how to endow our proposed NodeRENs with contractivity and dissipativity -- crucial properties for robust learning and control. Most importantly, as for RENs, we derive parametrizations of contractive and dissipative NodeRENs which are unconstrained, hence enabling their learning for a large number of parameters. We validate the properties of NodeRENs, including the possibility of handling irregularly sampled data, in a case study in nonlinear system identification.