# Discovering conservation laws from data for control

arXiv: https://arxiv.org/abs/1811.00961 · HTML: https://mlsanity.com/paper/1811.00961/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2018.

## Built on by (most cited first, 14 of 14)

- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Applying Machine Learning to Study Fluid Mechanics](https://mlsanity.com/paper/2110.02083/) (2021)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [Discovering conservation laws from trajectories via machine learning](https://mlsanity.com/paper/2102.04008/) (2021)
- [Weak Form Generalized Hamiltonian Learning](https://mlsanity.com/paper/2104.05096/) (2021)
- [Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation law](https://mlsanity.com/paper/2406.02154/) (2024)
- [Deep Learning of Conjugate Mappings](https://mlsanity.com/paper/2104.01874/) (2021)
- [Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation](https://mlsanity.com/paper/2410.05445/) (2024)
- [Discovering New Interpretable Conservation Laws as Sparse Invariants](https://mlsanity.com/paper/2305.19525/) (2023)
- [Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI](https://mlsanity.com/paper/2511.04564/) (2025)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)
- [When Do Conservation Laws Survive Learned Representations? Certified Horizons for Latent World Models](https://mlsanity.com/paper/2606.24945/) (2026)
- [Diagnosing Faults in Reinforcement Learning Simulators and World Models with Canonical Polynomial Invariants](https://mlsanity.com/paper/2609.13194/) (2026)

## Abstract (arXiv)

Conserved quantities, i.e. constants of motion, are critical for characterizing many dynamical systems in science and engineering. These quantities are related to underlying symmetries and they provide fundamental knowledge about physical laws, describe the evolution of the system, and enable system reduction. In this work, we formulate a data-driven architecture for discovering conserved quantities based on Koopman theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Interestingly, eigenfunctions of the Koopman operator associated with vanishing eigenvalues correspond to conserved quantities of the underlying system. In this paper, we show that these invariants may be identified with data-driven regression and power series expansions, based on the infinitesimal generator of the Koopman operator. We further establish a connection between the Koopman framework, conserved quantities, and the Lie-Poisson bracket. This data-driven method for discovering conserved quantities is demonstrated on the three-dimensional rigid body equations, where we simultaneously discover the total energy and angular momentum and use these intrinsic coordinates to develop a model predictive controller to track a given reference value.