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

TL;DR

NGCG is a neural-symbolic pipeline that reliably discovers conservation laws from data across diverse systems, effectively handling challenges like false positives, noise, and parameter variation, while providing interpretable closed-form expressions.

Contribution

The paper introduces NGCG, a novel neural-symbolic method that systematically improves conservation law discovery by decoupling dynamics learning from invariant extraction and eliminating false positives.

Findings

Achieves perfect detection (DR=1.0) on all systems with true laws.

Successfully discovers conservation laws in complex systems like Lotka-Volterra.

Robust to noise and sample-efficient, with runtime under one minute.

Abstract

Conservation laws are fundamental to understanding dynamical systems, but discovering them from data remains challenging due to parameter variation, non-polynomial invariants, local minima, and false positives on chaotic systems. We introduce NGCG, a neural-symbolic pipeline that decouples dynamics learning from invariant discovery and systematically addresses these challenges. A multi-restart variance minimiser learns a near-constant latent representation; system-specific symbolic extraction (polynomial Lasso, log-basis Lasso, explicit PDE candidates, and PySR) yields closed-form expressions; a strict constancy gate and diversity filter eliminate spurious laws. On a benchmark of nine diverse systems including Hamiltonian and dissipative ODEs, chaos, and PDEs, NGCG achieves consistent discovery (DR=1.0, FDR=0.0, F1=1.0) on all four systems with true conservation laws, with constancy two to three orders of magnitude lower than the best baseline. It is the only method that succeeds on the Lotka--Volterra system, and it correctly outputs no law on all five systems without invariants. Extensive experiments demonstrate robustness to noise ($\sigma = 0.1$), sample efficiency (50--100 trajectories), insensitivity to hyperparameters, and runtime under one minute per system. A Pareto analysis shows that the method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG achieves strong performance relative to prior methods for data-driven conservation-law discovery, combining high accuracy with interpretability.