🤖 AI Summary
This work addresses the challenge of discovering conservation laws from data in the presence of parameter variations, non-polynomial forms, local minima, and spurious correlations—particularly in chaotic systems. The authors propose NGCG, a neurosymbolic pipeline that decouples dynamics learning from invariant discovery: it first learns approximately constant latent representations by minimizing variance across multiple initializations, then employs diverse symbolic regression techniques to generate candidate expressions. Rigorous constancy gating and diversity-based filtering are introduced to eliminate false positives. NGCG achieves zero false discoveries for the first time, attaining perfect scores (DR = 1.0, FDR = 0.0, F1 = 1.0) across nine benchmark systems—including chaotic, dissipative, and partial differential equation models—with conservation-law constancy errors two to three orders of magnitude lower than the best baseline. The method also demonstrates robustness to noise, high sample efficiency, hyperparameter insensitivity, and runs in minutes per system.
📝 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 ($σ= 0.1$), 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.