Research map / Probabilistic and scientific modeling / Dynamical systems
Interpretable Meta-Learning of Physical Systems
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Proposes CAMEL, a context-affine multi-task representation learning model whose task weights are trained as meta-parameters, proving it identifies linearly parametrized physical parameters up to a linear transform, and shows fast, competitive multi-environment learning on physics systems.
Problem
Multi-environment learning of physical systems from few samples per environment, where existing meta-learning methods use black-box networks that are computationally costly and hard to interpret physically.
Approach
Model task variability as an affine function of low-dimensional task-specific weights stored as meta-parameters (no inner loop), so adaptation is least squares; under linear physical parameters, the learned weights identify the physical parameters via a linear map, enabling zero-shot adaptation (phi-CAMEL).
Key results
- CAMEL achieves 1.0e-4 adaptation MSE on point charges (10-shot) versus 8.2e-2 for CoDA and 9.2e-4 for ANIL, and identifies charges with under 1% relative error with 10 training tasks.
- On the cartpole at 100-shot, CAMEL reaches 3.1e-3 MSE versus 2.5e-2 for ANIL and 8.1e-1 for CoDA.
- On Upkie, CAMEL achieves 8.2e-3 adaptation error versus 1.5e-2 (MAML), 1.9e-2 (ANIL) and 2.1e-2 (CoDA), with gradient-free adaptation fast enough for a 200Hz MPC loop.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CAMEL instead of gradient-based meta-learning baselines | Multi-environment physical-system learning · Charges, Cartpole, Arm, Upkie | adaptation error: Charges 10-shot: 9.2e-4 ± 5e-4 -> 1.0e-4 ± 5e-5; Cartpole 100-shot: 2.5e-2 ± 9e-2 -> 3.1e-3 ± 5e-4; Arm 100-shot: 7.5e-1 | Helped |
| Use CAMEL instead of CoDA | Learning the ε-Capacitor · ε-Capacitor ε=0.1 | adaptation error: 1.0e-3 ± 5e-4 -> 1.9e-4 ± 2e-5 | Helped |
| Use CAMEL instead of CoDA | Capacitor field prediction · Capacitor, 40-shot | adaptation error: 2.6e-2± 1e-2 -> 2.6e-2 ± 1e-2 | No clear effect |
| Use CAMEL instead of ANIL | Learning the capacitor system · Capacitor, 5-shot | adaptation error: 4.4e-2 ± 2e-2 -> 3.6e-2 ± 2e-2 | No clear effect |
| Increase the number of inner gradient steps for MAML and ANIL | MAML and ANIL training · Physical-system experiments | training stability | Helped |
| Increase the number of inner gradient steps for MAML and ANIL | MAML and ANIL training · Physical-system experiments | training time | Hurt |
Surprising findings
- On the nonlinear capacitor, CAMEL and baselines perform similarly in accuracy, but CAMEL is much faster.
- ANIL and CoDA gradient-based adaptation was too slow to run in the 200Hz model predictive control loop on Upkie.
Limitations
- Identifiability guarantee relies on linear physical parameters; for nonlinear contexts only local identification holds.
- Number of training tasks is assumed to be limited and the identification map needs sufficient training tasks (e.g., 10 for accurate charge identification).
Experimental setup
Datasets: simulated point charge fields simulated capacitor fields (Poisson solver) simulated cartpole and acrobot trajectories with friction simulated Upkie balancing trajectories
Benchmarks: point charges capacitor epsilon-capacitor cartpole arm Upkie
Metrics: adaptation mean squared error relative identification error computational time
Baselines: MAML ANIL CoDA R2-D2
Builds on: MAML ANIL CoDA R2-D2
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