From Data to Laws: Neural Discovery of Conservation Laws Without False Positives

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 $t=0$ ($λ_{\mathrm{align}}=0.2$). With $λ_{\mathrm{align}}=0$, 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.

Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty. Here we introduce SymLang (Symmetry-constrained Language-guided equation discovery), a unified framework that brings together three previously separate ideas: (i) typed symmetry-constrained grammars that encode dimensional analysis, group-theoretic invariance, and parity constraints as hard production rules, eliminating on average 71.3% of candidate expression trees before any fitting; (ii) language-model-guided program synthesis in which a fine-tuned 7B-parameter proposer, conditioned on interpretable data descriptors, efficiently navigates the constrained search space; and (iii) MDL-regularized Bayesian model selection coupled with block-bootstrap stability analysis that quantifies structural uncertainty rather than committing to a single best equation. Across 133 dynamical systems spanning classical mechanics, electrodynamics, thermodynamics, population dynamics, and nonlinear oscillators, SymLang achieves an exact structural recovery rate of 83.7% under 10% observational noise - a 22.4 percentage-point improvement over the next-best baseline - while reducing out-of-distribution extrapolation error by 61% and near-eliminating conservation-law violations (3.1 x 10-3 vs. 187.3 x 10-3 physical drift for the closest competitor). In all tested regimes the framework correctly identifies structural degeneracy, reporting it explicitly rather than returning a confidently wrong single equation. The framework is fully open-source and reproducible, providing a principled pathway from raw data to interpretable, physically auditable symbolic laws.

Understanding natural and engineered systems often relies on symbolic formulations, such as differential equations, which provide interpretability and transferability beyond black-box models. We introduce Latent Grammar Flow (LGF), a neuro-symbolic generative framework for discovering ordinary differential equations from data. LGF embeds equations as grammar-based representations into a discrete latent space and forces semantically similar equations to be positioned closer together with a behavioural loss. Then, a discrete flow model guides the sampling process to recursively generate candidate equations that best fit the observed data. Domain knowledge and constraints, such as stability, can be either embedded into the rules or used as conditional predictors.