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Towards Cross Domain Generalization of Hamiltonian Representation via Meta Learning
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Meta-trains a GCN-based Hamiltonian network with MAML across six heterogeneous Hamiltonian systems so it can adapt to an unseen system type with few gradient steps.
Problem
Existing neural Hamiltonian and dynamics-learning methods are system-specific and generalize only across parameter environments of one system type, so they fail on systems with different functional-form Hamiltonians.
Approach
Apply MAML to a GCN Hamiltonian regressor trained over a distribution of different Hamiltonian system types (mass-spring, pendulum, Hénon-Heiles, magnetic-mirror, two-body, three-body), holding out one type for adaptation, and use CKA to argue the meta-trained model learns a shared Hamiltonian representation.
Key results
- Meta-trained model achieves lower relative error than pre-trained and randomly initialized (HNN) models on held-out mass-spring, pendulum, Hénon-Heiles, and magnetic-mirror systems within 50 adaptation steps.
- Meta-trained model shows substantially lower 1-CKA values in the last layer during adaptation across all four held-out systems than the other models.
- Meta-trained model outperforms CoDA and DyAd in the cross-domain setting, and adapts to unseen 2D harmonic oscillator and Kepler systems.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use meta-training instead of ordinary pre-training | GNN Hamiltonian model · Held-out Hamiltonian systems | relative error | Helped |
| Use meta-training instead of random initialization | GNN Hamiltonian model · Held-out Hamiltonian systems | adaptation speed | Helped |
| Use meta-training instead of pre-training or random initialization | GNN Hamiltonian model · Mass-spring | trajectory prediction quality | Mixed |
| Use meta-training instead of pre-training or random initialization | GNN Hamiltonian model · Pendulum | trajectory prediction quality | Helped |
| Use meta-training instead of pre-training or random initialization | GNN Hamiltonian model · Hénon-Heiles and magnetic-mirror | trajectory prediction quality | Helped |
| Use meta-training instead of CoDA | Cross-domain Hamiltonian dynamics model · Held-out Hamiltonian systems | trajectory prediction quality | Helped |
| Use the meta-trained model instead of DyAd | Magnetic-mirror adaptation · Magnetic-mirror | adaptation performance | Helped |
| Adapt a model meta-trained only on conservative systems to a dissipative system | Meta-trained Hamiltonian model · Damped mass-spring | adaptation performance | Hurt |
| Adapt the meta-trained model to additional held-out system types | Meta-trained Hamiltonian model · 2D harmonic oscillator and Kepler | relative error | Mixed |
| Use a symplectic integrator instead of LSODA (ablation) | Adapted Hamiltonian models · Integrator ablation across adaptation tasks | relative error | Hurt |
| Increase adaptation data size K (ablation) | Meta-trained model · Held-out adaptation tasks | relative error | Helped |
| Use meta-training for adaptation with low K (ablation) | Hamiltonian model · Held-out adaptation tasks | relative error | Helped |
Surprising findings
- Adaptation to a damped (dissipative) mass-spring system fails, showing the meta-trained conservative model does not transfer to non-conservative dynamics.
- For two-body and three-body systems none of the models successfully capture overall dynamics, with results described as preliminary.
Limitations
- Only conservative systems are used in meta-training; adaptation to dissipative systems fails.
- Two-body and three-body results are preliminary and the meta-learning configuration needs refinement for multi-particle tasks.
- Hyperparameters of the network architecture were not heavily tuned.
Experimental setup
Datasets: simulated trajectories for mass-spring, pendulum, Hénon-Heiles, magnetic-mirror, two-body, three-body systems
Benchmarks: mass-spring pendulum Hénon-Heiles magnetic-mirror two-body three-body 2D harmonic oscillator Kepler damped mass-spring
Metrics: relative error of predicted coordinates relative error of energy 1-CKA
Baselines: pre-trained model without meta learning randomly initialized model (HNN) CoDA DyAd
Builds on: MAML Hamiltonian Neural Network CoDA DyAd
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