From Data to Laws: Neural Discovery of Conservation Laws Without False Positives
arXiv:2603.20474v1 Announce Type: new 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 ba
arXiv:2603.20474v1 Announce Type: new 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.
Executive Summary
This article presents a novel approach to discovering conservation laws from data using a neural-symbolic pipeline called NGCG. NGCG decouples dynamics learning from invariant discovery, addressing challenges such as parameter variation, non-polynomial invariants, and false positives. The method achieves consistent discovery on a benchmark of nine diverse systems, including Hamiltonian and dissipative ODEs, chaos, and PDEs. NGCG demonstrates robustness to noise, sample efficiency, and insensitivity to hyperparameters, with a runtime under one minute per system. The method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG offers strong performance relative to prior methods, combining high accuracy with interpretability.
Key Points
- ▸ NGCG is a neural-symbolic pipeline that decouples dynamics learning from invariant discovery
- ▸ NGCG addresses challenges such as parameter variation, non-polynomial invariants, and false positives
- ▸ NGCG achieves consistent discovery on a benchmark of nine diverse systems
Merits
Strength in Addressing Challenges
NGCG effectively addresses the challenges of discovering conservation laws from data, including parameter variation, non-polynomial invariants, and false positives.
High Accuracy and Interpretability
NGCG achieves strong performance relative to prior methods, combining high accuracy with interpretability.
Demerits
Limited Domain Expertise
The article assumes a high level of domain knowledge in physics and mathematics, which may limit its accessibility to a broader audience.
Potential Overfitting
NGCG's reliance on neural networks and symbolic extraction may lead to overfitting, particularly when dealing with small datasets.
Expert Commentary
NGCG represents a significant advancement in the field of data-driven discovery of physical laws. The method's ability to decouple dynamics learning from invariant discovery and its robustness to noise and hyperparameters are notable strengths. However, the article's assumption of high domain expertise and potential for overfitting are limitations that should be addressed in future research. Furthermore, the development of NGCG highlights the importance of interdisciplinary research and collaboration between physicists, mathematicians, and computer scientists.
Recommendations
- ✓ Future research should focus on applying NGCG to real-world systems and addressing the limitations of the method.
- ✓ Developing more robust and interpretable methods for discovering conservation laws from data is essential for advancing the field of data-driven physics.
Sources
Original: arXiv - cs.LG
