# Learning Hamiltonian dynamics by reservoir computer

arXiv: https://arxiv.org/abs/2104.14474 · HTML: https://mlsanity.com/paper/2104.14474/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 24 Apr 2021.

## Builds on (most cited first, 13 of 13)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Machine learning prediction of critical transition and system collapse](https://mlsanity.com/paper/2012.01545/) (2020)
- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Model-free prediction of spatiotemporal dynamical systems with recurrent neural networks: Role of network spectral radius](https://mlsanity.com/paper/1910.04426/) (2019)
- [Anticipating synchronization with machine learning](https://mlsanity.com/paper/2103.13358/) (2021)
- [Forecasting Hamiltonian dynamics without canonical coordinates](https://mlsanity.com/paper/2010.15201/) (2020)
- [Dynamical learning of dynamics](https://mlsanity.com/paper/1902.02875/) (2019)
- [Toward an AI Physicist for Unsupervised Learning](https://mlsanity.com/paper/1810.10525/) (2018)
- [Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networks](https://mlsanity.com/paper/2001.00111/) (2019)
- [Inferring the dynamics of oscillatory systems using recurrent neural networks](https://mlsanity.com/paper/1904.03026/) (2019)
- [Transfer learning of chaotic systems](https://mlsanity.com/paper/2011.09970/) (2020)

## Built on by (most cited first, 2 of 2)

- [Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics](https://mlsanity.com/paper/2507.06050/) (2025)
- [Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach](https://mlsanity.com/paper/2604.00473/) (2026)

## Abstract (arXiv)

Reconstructing the KAM dynamics diagram of Hamiltonian system from the time series of a limited number of parameters is an outstanding question in nonlinear science, especially when the Hamiltonian governing the system dynamics are unknown. Here, we demonstrate that this question can be addressed by the machine learning approach knowing as reservoir computer (RC). Specifically, we show that without prior knowledge about the Hamilton's equations of motion, the trained RC is able to not only predict the short-term evolution of the system state, but also replicate the long-term ergodic properties of the system dynamics. Furthermore, by the architecture of parameter-aware RC, we also show that the RC trained by the time series acquired at a handful parameters is able to reconstruct the entire KAM dynamics diagram with a high precision by tuning a control parameter externally. The feasibility and efficiency of the learning techniques are demonstrated in two classical nonlinear Hamiltonian systems, namely the double-pendulum oscillator and the standard map. Our study indicates that, as a complex dynamical system, RC is able to learn from data the Hamiltonian.