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

TL;DR

This paper investigates whether neural networks can learn conserved quantities like energy in physical systems, comparing structured models, conservation discovery networks, and baselines across noisy and clean data.

Contribution

It demonstrates that specialized neural network architectures can accurately learn conserved quantities, and highlights the importance of temporal consistency and training data in this process.

Findings

Structured energy models achieve near-perfect $R^2$ on clean data.

Conservation Discovery Networks perform well with temporal consistency and alignment loss.

Noise robustness varies, with CDN outperforming structured models under certain noisy conditions.

Abstract

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 $R^2 \geq 0.9999$ against analytical energy on clean data, while the black-box CDN reaches $R^2 \geq 0.996$ when trained with temporal consistency plus a small alignment loss to analytical energy at $t=0$ ($\lambda_{\mathrm{align}}=0.2$). With $\lambda_{\mathrm{align}}=0$, CDN Pearson $R^2$ 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 $R^2=0.78$ under a short training schedule on the pendulum system, but reaches $R^2=0.9998$ with more training time and data, regardless of whether noise is added.