# Neural Contractive Dynamical Systems

arXiv: https://arxiv.org/abs/2401.09352 · HTML: https://mlsanity.com/paper/2401.09352/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 46 references that are in the map.
- First posted on arXiv 17 Jan 2024.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks](https://mlsanity.com/paper/2009.11848/) (2020)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows](https://mlsanity.com/paper/2010.13129/) (2020)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach](https://mlsanity.com/paper/2006.04361/) (2020)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [Learning Contracting Vector Fields For Stable Imitation Learning](https://mlsanity.com/paper/1804.04878/) (2018)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Avoiding Dense and Dynamic Obstacles in Enclosed Spaces: Application to Moving in Crowds](https://mlsanity.com/paper/2105.11743/) (2021)
- [Reaching Through Latent Space: From Joint Statistics to Path Planning in Manipulation](https://mlsanity.com/paper/2210.11779/) (2022)
- [Active Learning based on Data Uncertainty and Model Sensitivity](https://mlsanity.com/paper/1808.02026/) (2018)
- [RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks](https://mlsanity.com/paper/2106.08928/) (2021)
- [A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stability](https://mlsanity.com/paper/2110.00693/) (2021)

## Built on by (most cited first, 14 of 14)

- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Predictability Enables Parallelization of Nonlinear State Space Models](https://mlsanity.com/paper/2508.16817/) (2025)
- [Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recovery](https://mlsanity.com/paper/2412.07544/) (2024)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [Characterizing control between interacting subsystems with deep Jacobian estimation](https://mlsanity.com/paper/2507.01946/) (2025)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Denoising-based Contractive Imitation Learning](https://mlsanity.com/paper/2503.15918/) (2025)
- [TamedPUMA: safe and stable imitation learning with geometric fabrics](https://mlsanity.com/paper/2503.17432/) (2025)
- [Verifying Closed-Loop Contractivity of Learning-Based Controllers via Partitioning](https://mlsanity.com/paper/2512.02262/) (2025)
- [Stability Enhanced Gaussian Process Variational Autoencoders](https://mlsanity.com/paper/2604.09331/) (2026)
- [Let the Dynamics Flow: Stable Flow Matching Dynamical Systems](https://mlsanity.com/paper/2606.03834/) (2026)
- [Riemannian Simultaneous Inference for Tangent Vector Field Regression](https://mlsanity.com/paper/2609.21910/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)
- [Statistical Learning of Contractive Dynamical Representations for Composite Adaptive Control](https://mlsanity.com/paper/2609.35758/) (2026)

## Abstract (arXiv)

Stability guarantees are crucial when ensuring a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data, especially when the learned dynamics are governed by neural networks. We propose a novel methodology to learn neural contractive dynamical systems, where our neural architecture ensures contraction, and hence, global stability. To efficiently scale the method to high-dimensional dynamical systems, we develop a variant of the variational autoencoder that learns dynamics in a low-dimensional latent representation space while retaining contractive stability after decoding. We further extend our approach to learning contractive systems on the Lie group of rotations to account for full-pose end-effector dynamic motions. The result is the first highly flexible learning architecture that provides contractive stability guarantees with capability to perform obstacle avoidance. Empirically, we demonstrate that our approach encodes the desired dynamics more accurately than the current state-of-the-art, which provides less strong stability guarantees.