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Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees
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Proposes ELCD, a neural dynamical system parametrized via extended linearization that is globally exponentially stable and globally contracting in some metric, with a learned diffeomorphism to a latent space for arbitrary metrics.
Problem
Learning neural dynamical systems from demonstrations with global stability and contraction (robustness) guarantees, rather than only local contractivity near training data.
Approach
Parametrize the vector field as f(x)=A(x,x*)(x-x*) with the symmetric part of A forced negative definite (A = -PsᵀPs + Pa - Paᵀ - αI), guaranteeing global exponential stability and contraction in some metric via a converse contraction theorem; jointly train a coupling-layer diffeomorphism to a same-dimensional latent space so contraction holds in arbitrary data-space metrics.
Key results
- ELCD achieves the lowest mean DTWD on all eight settings, e.g., LASA-2D 0.12±0.11 vs NCDS 0.59±0.61 and SDD 0.37±0.32.
- On Pendulum-4D ELCD reaches 0.03±0.01 versus 0.17 (EFlow) and 1.35 (NCDS).
- SDD produced NaN on Rosenbrock, while ELCD reached 1.22±0.01 on Rosenbrock-8D vs EFlow 1.90±0.16.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use ELCD instead of the strongest baseline (SDD) | LASA-2D trajectory learning · LASA-2D | DTWD: 0.37 $\pm$ 0.32 -> 0.12 $\pm$0.11 | Helped |
| use ELCD instead of the strongest baseline (NCDS) | LASA-4D trajectory learning · LASA-4D | DTWD: 2.19 $\pm$ 1.23 -> 0.80 $\pm$ 0.54 | Helped |
| use ELCD instead of the strongest baseline (EFlow) | LASA-8D trajectory learning · LASA-8D | DTWD: 2.66 $\pm$ 0.63 -> 1.52 $\pm$ 0.61 | Helped |
| use ELCD instead of the strongest baseline (EFlow) | Pendulum-4D trajectory learning · Pendulum-4D | DTWD: 0.17 $\pm$ 0.01 -> 0.03 $\pm$ 0.01 | Helped |
| use ELCD instead of the strongest baseline (EFlow) | Pendulum-8D trajectory learning · Pendulum-8D | DTWD: 0.33 $\pm$ 0.01 -> 0.14 $\pm$ 0.03 | Helped |
| use ELCD instead of the strongest baseline (EFlow) | Pendulum-16D trajectory learning · Pendulum-16D | DTWD: 0.45 $\pm$ 0.01 -> 0.44 $\pm$ 0.09 | Helped |
| use ELCD instead of the strongest baseline (EFlow) | Rosenbrock-8D trajectory learning · Rosenbrock-8D | DTWD: 1.90 $\pm$ 0.16 -> 1.22 $\pm$ 0.01 | Helped |
| use ELCD instead of the strongest baseline (EFlow) | Rosenbrock-16D trajectory learning · Rosenbrock-16D | DTWD: 3.57 $\pm$ 0.66 -> 2.57 $\pm$ 0.09 | Helped |
| add a learned coordinate transform and train it jointly with ELCD | ELCD on trajectories from a system contracting in a general metric · Linear-system toy example | trajectory fit | Helped |
| train to predict velocity directly instead of predicting the next state with one Euler step | NCDS implementation · All experiments | performance | Helped |
Surprising findings
- Training a VAE-based coordinate transform separately before the dynamics (as in NCDS) failed to reasonably fit the data in the authors' implementation, motivating joint training.
- Stable Deep Dynamics diverged (NaN) on the Rosenbrock dataset.
Limitations
- Assumes the underlying data dynamics are contracting and converge to a single fixed point; no multiple equilibria, limit cycles or chaos.
- Currently limited to dynamics on R^n rather than general manifolds.
- Struggles to fit stiff, multi-timescale dynamics such as Rosenbrock (still open challenge).
Experimental setup
Datasets: LASA multi-link pendulum (simulated) Rosenbrock generalization (simulated)
Benchmarks: LASA multi-link pendulum Rosenbrock
Metrics: normalized dynamic time warping distance (DTWD) standard deviation across 10 runs
Baselines: NCDS Stable Deep Dynamics (SDD) Euclideanizing Flow (EFlow)
Builds on: neural contractive dynamical systems partial contraction theory stable deep dynamics euclideanizing flows mean-value theorem extended linearization
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