Prediction Is Not Physics: Learning and Evaluating Conserved Quantities in Neural Simulators

A diffusion model trained on Hamiltonian trajectories can achieve rollout MSE near 10-3, but the standard deviation of its energy over time is between 7500 and 36000 times larger than the ground-truth energy standard deviation, indicating a failure to preserve conservation laws. This gap motivates our central question of whether neural networks can learn or select globally conserved quantities from physical trajectories. We investigate this across three Hamiltonian systems: projectile motion, pendulum, and spring-mass. We use a structured T(v)+V(q) energy model, a black-box Conservation Discovery Network (CDN), a polynomial CDN, and a conditional diffusion baseline. The structured network reaches R2 ≥ 0.9999 against analytical energy on clean data, while the black-box CDN reaches R2 ≥ 0.996 when trained with temporal consistency plus a small alignment loss to analytical energy at t=0 (λalign=0.2). With λalign=0, CDN Pearson R2 collapses on pendulum and spring-mass (< 10-3), showing that temporal consistency alone is not enough to reliably identify the true energy. Under 1% additive Gaussian noise, the CDN outperforms the structured model on the projectile and spring-mass systems, suggesting that the CDN may be more robust to noisy inputs in this setting. However, the polynomial CDN is sensitive to training configuration: it achieves R2=0.78 under a short training schedule on the pendulum system, but reaches R2=0.9998 with more training time and data, regardless of whether noise is added.