# Discovering conservation laws from trajectories via machine learning

arXiv: https://arxiv.org/abs/2102.04008 · HTML: https://mlsanity.com/paper/2102.04008/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Self-training with Noisy Student improves ImageNet classification](https://mlsanity.com/paper/1911.04252/) (2019)
- [Deep Learning Face Representation by Joint Identification-Verification](https://mlsanity.com/paper/1406.4773/) (2014)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Improving Robustness Without Sacrificing Accuracy with Patch Gaussian Augmentation](https://mlsanity.com/paper/1906.02611/) (2019)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networks](https://mlsanity.com/paper/2003.04299/) (2020)
- [Machine Learning Topological Invariants with Neural Networks](https://mlsanity.com/paper/1708.09401/) (2017)
- [Discovering conservation laws from data for control](https://mlsanity.com/paper/1811.00961/) (2018)
- [Toward an AI Physicist for Unsupervised Learning](https://mlsanity.com/paper/1810.10525/) (2018)
- [Deep Learning Topological Invariants of Band Insulators](https://mlsanity.com/paper/1805.10503/) (2018)
- [Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networks](https://mlsanity.com/paper/2001.00111/) (2019)
- [Learning a Local Symmetry with Neural-Networks](https://mlsanity.com/paper/1904.07637/) (2019)

## Built on by (most cited first, 8 of 8)

- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks](https://mlsanity.com/paper/2304.12865/) (2023)
- [ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction](https://mlsanity.com/paper/2302.05783/) (2023)
- [Machine learning for knowledge acquisition and accelerated inverse-design for non-Hermitian systems](https://mlsanity.com/paper/2204.13376/) (2022)
- [Learning quantum symmetries with interactive quantum-classical variational algorithms](https://mlsanity.com/paper/2206.11970/) (2022)
- [Model-agnostic machine learning of conservation laws from data](https://mlsanity.com/paper/2301.07503/) (2023)
- [Understanding Learning through the Lens of Dynamical Invariants](https://mlsanity.com/paper/2401.10428/) (2024)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)

## Abstract (arXiv)

Invariants and conservation laws convey critical information about the underlying dynamics of a system, yet it is generally infeasible to find them from large-scale data without any prior knowledge or human insight. We propose ConservNet to achieve this goal, a neural network that spontaneously discovers a conserved quantity from grouped data where the members of each group share invariants, similar to a general experimental setting where trajectories from different trials are observed. As a neural network trained with a novel and intuitive loss function called noise-variance loss, ConservNet learns the hidden invariants in each group of multi-dimensional observables in a data-driven, end-to-end manner. Our model successfully discovers underlying invariants from the simulated systems having invariants as well as a real-world double pendulum trajectory. Since the model is robust to various noises and data conditions compared to baseline, our approach is directly applicable to experimental data for discovering hidden conservation laws and further, general relationships between variables.