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

arXiv: https://arxiv.org/abs/2603.20474 · HTML: https://mlsanity.com/paper/2603.20474/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 53 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 29 of 29)

- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data](https://mlsanity.com/paper/2012.02334/) (2020)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [Lie Point Symmetry and Physics Informed Networks](https://mlsanity.com/paper/2311.04293/) (2023)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Unravelling the Performance of Physics-informed Graph Neural Networks for Dynamical Systems](https://mlsanity.com/paper/2211.05520/) (2022)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [AI Poincar'e 2.0: Machine Learning Conservation Laws from Differential Equations](https://mlsanity.com/paper/2203.12610/) (2022)
- [A Review of Data-Driven Discovery for Dynamic Systems](https://mlsanity.com/paper/2210.10663/) (2022)
- [Discovering conservation laws from data for control](https://mlsanity.com/paper/1811.00961/) (2018)
- [Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2206.09299/) (2022)
- [Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2401.01783/) (2024)
- [Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networks](https://mlsanity.com/paper/2001.00111/) (2019)
- [Neural equilibria for long-term prediction of nonlinear conservation laws](https://mlsanity.com/paper/2501.06933/) (2025)
- [Discovering conservation laws from trajectories via machine learning](https://mlsanity.com/paper/2102.04008/) (2021)
- [ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction](https://mlsanity.com/paper/2302.05783/) (2023)
- [Discovering Physics Laws of Dynamical Systems via Invariant Function Learning](https://mlsanity.com/paper/2502.04495/) (2025)
- [Discovering interpretable Lagrangian of dynamical systems from data](https://mlsanity.com/paper/2302.04400/) (2023)
- [Interpretability in Symbolic Regression: a benchmark of Explanatory Methods using the Feynman data set](https://mlsanity.com/paper/2404.05908/) (2024)
- [Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation](https://mlsanity.com/paper/2410.05445/) (2024)
- [GeoHNNs: Geometric Hamiltonian Neural Networks](https://mlsanity.com/paper/2507.15678/) (2025)
- [Automated Discovery of Conservation Laws via Hybrid Neural ODE-Transformers](https://mlsanity.com/paper/2511.00102/) (2025)

## Abstract (arXiv)

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.