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arXiv · 2603.20474

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

Rahul D Ray
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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.

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Rahul D Ray. 2026-03-20. From Data to Laws: Neural Discovery of Conservation Laws Without False Positives. https://arxiv.org/abs/2603.20474

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