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Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees

Sean Jaffe, Alexander Davydov, Deniz Lapsekili, Ambuj Singh, Francesco Bullo

TL;DR

The paper tackles the challenge of learning neural dynamical systems with global stability and robustness guarantees in arbitrary metrics. It introduces Extended Linearized Contracting Dynamics (ELCD), which factorizes the vector field as $f(x)=A(x,x^*)(x-x^*)$ with a neural-parameterized $A$ whose symmetric part is negative definite, yielding global exponential stability and equilibrium contraction; it also learns a latent-space diffeomorphism of equal dimension to realize contraction in the data space under general metrics, trained jointly with the dynamics. The authors demonstrate that the joint training and the skew-symmetric component of $A$ enable accurate trajectory reproduction on high-dimensional LASA data, multi-link pendulum, and generalized Rosenbrock dynamics, outperforming baselines like NCDS, EFlow, and SDD. This work provides scalable, global contraction guarantees for neural contracting dynamics, with practical implications for robust imitation, control, and robotics.

Abstract

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.

Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees

TL;DR

The paper tackles the challenge of learning neural dynamical systems with global stability and robustness guarantees in arbitrary metrics. It introduces Extended Linearized Contracting Dynamics (ELCD), which factorizes the vector field as with a neural-parameterized whose symmetric part is negative definite, yielding global exponential stability and equilibrium contraction; it also learns a latent-space diffeomorphism of equal dimension to realize contraction in the data space under general metrics, trained jointly with the dynamics. The authors demonstrate that the joint training and the skew-symmetric component of enable accurate trajectory reproduction on high-dimensional LASA data, multi-link pendulum, and generalized Rosenbrock dynamics, outperforming baselines like NCDS, EFlow, and SDD. This work provides scalable, global contraction guarantees for neural contracting dynamics, with practical implications for robust imitation, control, and robotics.

Abstract

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.
Paper Structure (25 sections, 3 theorems, 27 equations, 4 figures, 1 table)

This paper contains 25 sections, 3 theorems, 27 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Preliminaries
  5. Contracting Linear Systems
  6. Contractivity and Exponential Stability for Nonlinear Systems
  7. Methods
  8. Motivation
  9. Parametrization of $A(x,x^*)$
  10. Latent Space Representation
  11. On the Interdependence of Diffeomorphism and Learned Dynamics
  12. Choice of Diffeomorphism
  13. Training
  14. Experiments
  15. Datasets
  16. ...and 10 more sections

Key Result

Lemma 2

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}^d$ be continuously differentiable. Then for every $x,y \in \mathbb{R}^d$,

Figures (4)

  • Figure 1: The learned vector field and corresponding trajectories of an ELCD with no transform (left) and a model with a transform (right) when trained on data that is generated from a vector field that is contracting in a more general metric.
  • Figure 2: Plots of the vector fields and induced trajectories learned by ELCD (Top) and NCDS (Bottom) after different training epochs. ELCD is contracting always while NCDS may admit multiple equilibrium or diverge.
  • Figure 3: Plots of the 2D LASA data. Demonstrations are in black. The learned ELCD trajectories in magenta are plotted along with the learned vector field. The vector field velocities have been normalized for visibility.
  • Figure 4: Phase plots of the two link-pendulum trajectories (blue) and the trajectories produced by ELCD (red). Each column is a different trajectory. The top row is the first link and the bottom row is the second link. The x-axis is angle and the y-axis is angular velocity.

Theorems & Definitions (5)

  • Definition 1
  • Lemma 2: Mean-value theorem (Proposition 2.4.7 in RA-JEM-TSR:88)
  • Theorem 3: Equilibrium Contraction and Global Exponential Stability
  • proof
  • Proposition 4: Theorem 2.8 in PG-SH-IM:21