# Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networks

arXiv: https://arxiv.org/abs/2001.00111 · HTML: https://mlsanity.com/paper/2001.00111/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2019.

## Built on by (most cited first, 10 of 10)

- [AI Poincar'e 2.0: Machine Learning Conservation Laws from Differential Equations](https://mlsanity.com/paper/2203.12610/) (2022)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Discovering conservation laws from trajectories via machine learning](https://mlsanity.com/paper/2102.04008/) (2021)
- [Learning Hamiltonian dynamics by reservoir computer](https://mlsanity.com/paper/2104.14474/) (2021)
- [Convolutional Autoencoders for Reduced-Order Modeling](https://mlsanity.com/paper/2108.12453/) (2021)
- [Learning quantum symmetries with interactive quantum-classical variational algorithms](https://mlsanity.com/paper/2206.11970/) (2022)
- [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)
- [Analysis of the Identifying Regulation with Adversarial Surrogates Algorithm](https://mlsanity.com/paper/2405.02953/) (2024)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)

## Abstract (arXiv)

Understanding complex systems with their reduced model is one of the central roles in scientific activities. Although physics has greatly been developed with the physical insights of physicists, it is sometimes challenging to build a reduced model of such complex systems on the basis of insights alone. We propose a novel framework that can infer the hidden conservation laws of a complex system from deep neural networks (DNNs) that have been trained with physical data of the system. The purpose of the proposed framework is not to analyze physical data with deep learning, but to extract interpretable physical information from trained DNNs. With Noether's theorem and by an efficient sampling method, the proposed framework infers conservation laws by extracting symmetries of dynamics from trained DNNs. The proposed framework is developed by deriving the relationship between a manifold structure of time-series dataset and the necessary conditions for Noether's theorem. The feasibility of the proposed framework has been verified in some primitive cases for which the conservation law is well known. We also apply the proposed framework to conservation law estimation for a more practical case that is a large-scale collective motion system in the metastable state, and we obtain a result consistent with that of a previous study.