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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
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Proposes Geometric Contact Flows (GCF), which learn dynamical systems by mapping a latent contact Hamiltonian system to the data space through an ensemble of contactomorphisms, with uncertainty-aware Riemannian geodesics for generalization and obstacle avoidance.
Problem
Learning dynamical systems with force exchange and dissipation that generalize reliably outside the training data support, while preserving physical structure and remaining interpretable.
Approach
Use a latent contact Hamiltonian model as an inductive bias and adapt it to target dynamics with trainable contactomorphisms (analytically invertible, built from contact splitting integrators). An ensemble of contactomorphisms gives predictive uncertainty, which reshapes the Riemannian (Jacobi-type) metric so geodesics avoid uncertain or unsafe regions.
Key results
- GCF reduces spring mesh reconstruction DTWD to 0.50 from 1.24 (DHNN) and 1.49 (HNN), a 57% reduction in error.
- On the quantum system, GCF reaches DTWD 0.29 versus 0.41 for MLP and 0.70–0.72 for NCDS and EF, a 60% reduction.
- On DigiLeTs (Elle), GCF achieves DTWD 0.70 versus 2.22 for EF and 2.79 for NCDS; convergence ratio on Elle is 0.66 versus 0.61 for EF.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use GCF instead of the strongest listed baseline | Spring mesh dynamics reconstruction · Spring mesh | DTWD: 0.50_{\pm .19} vs 1.24_{\pm .62} | Helped |
| use GCF instead of MLP | Quantum dynamics reconstruction · Quantum system | DTWD: 0.29_{\pm 0.04} vs 0.41_{\pm 0.06} | Helped |
| use GCF instead of EF | Handwriting reconstruction · LASA Leaf_2 | DTWD: 0.43_{\pm 0.10} vs 0.44_{\pm 0.12} | No clear effect |
| use GCF instead of DHNN | Handwriting reconstruction · DigiLeTs Elle | DTWD: 0.82_{\pm 0.22} vs 0.70_{\pm 0.36} | Helped |
| use GCF instead of NCDS | Generalization of handwriting dynamics · LASA Leaf_2 | convergence ratio: 0.94_{\pm 0.19} vs 0.69_{\pm 0.19} | Hurt |
| use GCF instead of NCDS | Generalization of handwriting dynamics · DigiLeTs Elle | convergence ratio: 0.54_{\pm 0.35} vs 0.66_{\pm 0.12} | Helped |
| use GCF instead of EF | Wrap-and-Pull reproduction · Wrap-and-Pull | DTWD: 1.79_{\pm 0.04} vs 0.61_{\pm 0.25} | Helped |
| use stable rather than safe latent dynamics | GCF · Wrap-and-Pull loaded | energy consumption (J): 0.63_{\pm .02} vs 0.81_{\pm .09} | Mixed |
Surprising findings
- Safe GCF variant halts motion when the Lagrangian action approaches zero, acting as an energy-based safety stop under an unexpected load.
Limitations
- Contact-geometry model only validated on a few physical, handwriting and robot tasks; the paper does not test large-scale or high-dimensional real-world systems beyond the 60-dimensional spring mesh.
- The Lagrangian action s is not directly observable and must be estimated via consistency between Maupertuis' principle and Noether's theorem.
Experimental setup
Datasets: spring mesh dataset LASA DigiLeTs
Benchmarks: LASA DigiLeTs
Metrics: dynamic time warping distance convergence ratio
Baselines: Euclideanizing Flows (EF) Neural Contractive Dynamical Systems (NCDS) Hamiltonian Neural Networks (HNN) Dissipative Hamiltonian Neural Networks (DHNN) MLP
Builds on: Euclideanizing Flows Neural Contractive Dynamical Systems Hamiltonian Neural Networks Dissipative Hamiltonian Neural Networks Maupertuis' principle
Closest papers by meaning
- Support-Safe Variational Hybrid Filtering for Contact-Mode and Sparse-Law Recovery (NeurIPS 2026 · similarity 0.85)
- ForceFlow: Learning to Feel and Act via Contact-Driven Flow Matching (NeurIPS 2026 · similarity 0.84)
- A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics (ICLR 2025 · similarity 0.84)
- Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures (ICML 2025 · similarity 0.84)
- Hamiltonian Score Matching and Generative Flows (NeurIPS 2024 · similarity 0.84)
- Learning Flexible Body Collision Dynamics with Hierarchical Contact Mesh Transformer (ICLR 2024 · similarity 0.84)
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