Trajectory-Consistent Dropout for Uncertainty Decomposition in Hamiltonian Neural Networks

Hamiltonian Neural Networks (HNNs) produce stable long-horizon rollouts by integrating a learned Hamiltonian with a symplectic scheme. Standard Monte Carlo (MC) dropout conflicts with this per-trajectory interpretation because resampling the dropout mask inside a leapfrog integrator means successive substeps follow different subnetworks, so a single MC rollout no longer corresponds to any one sampled Hamiltonian. We propose trajectory-consistent dropout, implemented as Fixed-mask MC, where each stochastic rollout selects one mask from a finite bank and reuses it across the full integration path. This restores a coherent per-rollout Hamiltonian sample and makes uncertainty decomposition operational. Crossing the mask axis with an initial-condition ensemble separates model-sample uncertainty from sensitivity to the initial state. On coupled-pendulum rollouts, Fixed-mask MC reduces mean energy drift by 72% relative to Standard MC. Under held-out mass ratios, Fixed-mask MC increases the epistemic share of predictive variance by 16.9 percentage points from in-distribution to out-of-distribution, compared with 4.2 percentage points for Standard MC, indicating a substantially stronger shift of the uncertainty budget toward model uncertainty when the physics moves outside training support. Spring-chain and transverse-field Ising experiments show that the same axis-preserving decomposition transfers to structured pairwise and latent quantum dynamics, although point-prediction and likelihood gains remain system-dependent. These results position Fixed-mask MC as a single-training-run route to trajectory-consistent, physically interpretable uncertainty in structured neural dynamical models.