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Neural Contractive Dynamical Systems
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Proposes Neural Contractive Dynamical Systems (NCDS), a neural architecture guaranteed contractive for all parameters via a negative-definite Jacobian integrated along line paths, combined with an injective VAE for latent contractive dynamics and SO(3) extension for robot motion learning.
Problem
Learning dynamical systems from demonstrations with global stability (contraction) guarantees, especially when the dynamics are parameterized by high-capacity neural networks and must scale to high-dimensional robot data.
Approach
Define the vector field's Jacobian as a negative-definite matrix -(J^T J + eps I) from an unconstrained network, integrate it along line paths to obtain f, so contraction holds by construction for any parameters; wrap it in an injective VAE (zero-padding flow decoder) so the decoded high-dimensional system remains contractive, and map rotations via Exp/Log on the first cover of so(3).
Key results
- On LASA-8D, NCDS achieves DTWD 2.28±0.24 vs. 4.69±0.52 for Imitation flow and 10.22±0.40 for Euclideanizing flow.
- On the 7-DoF robot dataset, NCDS achieves DTWD 1.18±0.16 vs. 2.63±0.37 for Imitation flow and 2.69±0.18 for SEDS.
- On LASA-2D, Euclideanizing flow (0.72) and Imitation flow (0.80) outperform NCDS (1.37), though all methods perform comparably; only NCDS's nearby-trajectory distance decreases monotonically.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NCDS instead of Euclideanizing flow, Imitation flow, or SEDS | NCDS · LASA-2D | DTWD: 0.72\pm 0.12; 0.80\pm 0.24; 1.60\pm 0.44 -> 1.37\pm 0.40 | Hurt |
| Use NCDS instead of Euclideanizing flow or SEDS | NCDS · LASA-4D | DTWD: 3.23\pm 0.34; 3.08\pm 0.20 -> 0.98\pm 0.15 | Helped |
| Use NCDS instead of Euclideanizing flow, Imitation flow, or SEDS | NCDS · LASA-8D | DTWD: 10.22\pm 0.40; 4.69\pm 0.52; 4.85\pm 1.64 -> 2.28\pm 0.24 | Helped |
| Use NCDS instead of Euclideanizing flow, Imitation flow, or SEDS | NCDS · 7 DoF robot | DTWD: 5.11 \pm 0.30; 2.63 \pm 0.37; 2.69\pm 0.18 -> 1.18 \pm 0.16 | Helped |
| Use a latent-space full pipeline rather than learning directly in input space | NCDS · 8D | execution time (ms): 40.9 -> 20.2 | Helped |
| Remove the negative-definiteness constraint on the Jacobian (ablation) | NCDS · LASA-2D | contraction stability | Hurt |
| Use a larger regularization term epsilon (ablation) | NCDS · Reconstruction | accuracy | Hurt |
| Use Tanh or Softplus instead of Sigmoid activation (ablation) | NCDS · LASA trajectories | generalization | Mixed |
| Change the numerical integration time steps (ablation) | NCDS · Sine-wave toy example | reconstruction accuracy | No clear effect |
Surprising findings
- On the 2D LASA task, NCDS is less accurate than Euclideanizing and Imitation flows in DTWD, despite being the only contractive method.
- Unconstrained MLP and NeuralODE models fit the vector field well but do not achieve contraction stability.
Limitations
- Sensitive to numerical integration scheme; requires adaptive-step integrators such as dopri5, whose variable run time may hurt real-time control.
- Numerical integration increases running time relative to standard feedforward networks.
- Obstacle avoidance is tailored to end-effector level; multi-limb obstacle avoidance requires refinement of the modulation matrix and joint-space training.
Experimental setup
Datasets: LASA Franka Emika Panda joint-space demonstrations (5 trajectories) KIT human motion dataset (Tennis forehand)
Benchmarks: LASA-2D LASA-4D LASA-8D 7-DoF Franka Emika robot dataset
Metrics: dynamic time warping distance (DTWD) average distance between nearby path integrals over time
Baselines: Euclideanizing flow Imitation flow SEDS MLP NeuralODE
Builds on: contraction theory Euclideanizing flow Imitation flow SEDS M-flows neural ODE
Closest papers by meaning
- Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees (NeurIPS 2024 · similarity 0.90)
- Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recovery (ICLR 2025 · similarity 0.87)
- Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space (NeurIPS 2024 · similarity 0.86)
- Learning Deep Dissipative Dynamics (AAAI 2025 · similarity 0.86)
- Stable Port-Hamiltonian Neural Networks (NeurIPS 2025 · similarity 0.85)
- Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation (ICML 2024 · similarity 0.85)
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