# Learning Symmetries of Classical Integrable Systems

arXiv: https://arxiv.org/abs/1906.04645 · HTML: https://mlsanity.com/paper/1906.04645/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2019.

## Built on by (most cited first, 15 of 15)

- [Generalized Lagrangian Neural Networks](https://mlsanity.com/paper/2401.03728/) (2024)
- [The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learning](https://mlsanity.com/paper/2103.04913/) (2021)
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- [Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation](https://mlsanity.com/paper/2410.05445/) (2024)
- [KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss](https://mlsanity.com/paper/2102.11923/) (2021)
- [Learning Integrable Dynamics with Action-Angle Networks](https://mlsanity.com/paper/2211.15338/) (2022)
- [Learning finite symmetry groups of dynamical systems via equivariance detection](https://mlsanity.com/paper/2503.03014/) (2025)
- [Discovering Symmetries of ODEs by Symbolic Regression](https://mlsanity.com/paper/2506.19550/) (2025)
- [Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI](https://mlsanity.com/paper/2511.04564/) (2025)
- [A charge-preserving method for solving graph neural diffusion networks](https://mlsanity.com/paper/2312.10279/) (2023)
- [Deep Learning based discovery of Integrable Systems](https://mlsanity.com/paper/2503.10469/) (2025)
- [Attention-based optimizer for symmetry finding](https://mlsanity.com/paper/2605.30429/) (2026)

## Abstract (arXiv)

The solution of problems in physics is often facilitated by a change of variables. In this work we present neural transformations to learn symmetries of Hamiltonian mechanical systems. Maintaining the Hamiltonian structure requires novel network architectures that parametrize symplectic transformations. We demonstrate the utility of these architectures by learning the structure of integrable models. Our work exemplifies the adaptation of neural transformations to a family constrained by more than the condition of invertibility, which we expect to be a common feature of applications of these methods.