# Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees

arXiv: https://arxiv.org/abs/2402.08090 · HTML: https://mlsanity.com/paper/2402.08090/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 50 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 25 of 25)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (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)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [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)
- [Learning Contracting Vector Fields For Stable Imitation Learning](https://mlsanity.com/paper/1804.04878/) (2018)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Beyond Convexity -- Contraction and Global Convergence of Gradient Descent](https://mlsanity.com/paper/1806.06655/) (2018)
- [High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds](https://mlsanity.com/paper/2010.10904/) (2020)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Non-Euclidean Contraction Theory for Robust Nonlinear Stability](https://mlsanity.com/paper/2103.12263/) (2021)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Learning Control-Oriented Dynamical Structure from Data](https://mlsanity.com/paper/2302.02529/) (2023)
- [Dual Seminorms, Ergodic Coefficients and Semicontraction Theory](https://mlsanity.com/paper/2201.03103/) (2022)
- [k-Contraction: Theory and Applications](https://mlsanity.com/paper/2008.10321/) (2020)
- [Generalization of the multiplicative and additive compounds of square matrices and contraction in the Hausdorff dimension](https://mlsanity.com/paper/2012.13441/) (2020)

## Built on by (most cited first, 11 of 11)

- [Predictability Enables Parallelization of Nonlinear State Space Models](https://mlsanity.com/paper/2508.16817/) (2025)
- [R2DN: Scalable Parameterization of Contracting and Lipschitz Recurrent Deep Networks](https://mlsanity.com/paper/2504.01250/) (2025)
- [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)
- [Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions](https://mlsanity.com/paper/2411.11405/) (2024)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Tracking control of latent dynamic systems with application to spacecraft attitude control](https://mlsanity.com/paper/2412.06342/) (2024)
- [Stable neural networks and connections to continuous dynamical systems](https://mlsanity.com/paper/2510.22299/) (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)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)

## Abstract (arXiv)

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets.