Rf Hgn Eval

rf-hgn-eval

Rapid training of Hamiltonian graph networks using random features — Rahma et al. (2025) (arXiv:2506.06558, 2025)

What this evaluates

Evaluates the training efficiency, accuracy, and zero-shot generalization capability of Hamiltonian Graph Networks (RF-HGNs) on mass-spring physical systems. It benchmarks the proposed random-feature training method against standard gradient-based optimizers and existing physics-informed graph architectures.

Datasets

• 3D lattice mass-spring system — total ?; splits: train (-1), test (-1)
• 2D open chain mass-spring system — total ?; splits: train (-1), test (-1)
• 2D closed chain mass-spring system (Thangamuthu et al. [87]) — total ?; splits: train (-1), test (-1)

Metrics

• Test MSE (primary) — range: other
  • Mean squared error between the true and predicted target values (e.g., Hamiltonian or trajectories).
• relative error — range: other
  • L2 norm of the difference between true and predicted values divided by the L2 norm of the true values: ||x_true - x_pred||_2 / ||x_true||_2.
• Train time [s] — range: seconds
  • Wall-clock time required to complete the training procedure.
• Speed-up — range: ratio
  • Ratio of training time of a baseline optimizer to the training time of RF-HGN.

Input / output format

Input: Graph structures representing mass-spring systems, where nodes contain position (q), momentum (p), mass (alpha), and spring constants (beta) in 2D or 3D space.

Output: Predicted Hamiltonian values or system trajectories corresponding to the input graph configurations.

Scoring recipe

def compute_metrics(x_true, x_pred):
    mse = np.mean((x_true - x_pred) ** 2)
    rel_err = np.linalg.norm(x_true - x_pred) / np.linalg.norm(x_true)
    return {'Test MSE': mse, 'relative error': rel_err}

Common pitfalls

• Early stopping was applied to all iterative optimizers, which may conflate convergence speed with final accuracy.
• Hyperparameters were tuned separately for each optimizer rather than using a unified search space, potentially biasing the efficiency comparison.
• Zero-shot generalization fails on 2x2 training systems because they lack nodes with four edges, which are present in larger test grids.

Evidence (verbatim from paper)

For $x_{\mathrm{true}}$, $x_{\mathrm{pred}} \in \mathbb{R}^m$ for $m \in \mathbb{N}$, we define the relative error as $||x_{\mathrm{true}} - x_{\mathrm{pred}}||2 / ||x{\mathrm{true}}||_2$.

Citation

@misc{rahma2025rapid,
  title={Rapid training of Hamiltonian graph networks using random features},
  author={Rahma et al. (2025)},
  year={2025},
  note={arXiv:2506.06558}
}

• arXiv: 2506.06558