# Physics-Informed Echo State Networks

arXiv: https://arxiv.org/abs/2011.02280 · HTML: https://mlsanity.com/paper/2011.02280/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2020.

## Built on by (most cited first, 16 of 16)

- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Analogue and Physical Reservoir Computing Using Water Waves](https://mlsanity.com/paper/2306.09095/) (2023)
- [CD-ROM: Complemented Deep-Reduced Order Model](https://mlsanity.com/paper/2202.10746/) (2022)
- [Frame-independent vector-cloud neural network for nonlocal constitutive modeling on arbitrary grids](https://mlsanity.com/paper/2103.06685/) (2021)
- [Data-driven prediction and control of extreme events in a chaotic flow](https://mlsanity.com/paper/2204.11682/) (2022)
- [Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks](https://mlsanity.com/paper/2304.12865/) (2023)
- [Physics-Informed Long Short-Term Memory for Forecasting and Reconstruction of Chaos](https://mlsanity.com/paper/2302.10779/) (2023)
- [Simultaneous boundary shape estimation and velocity field de-noising in Magnetic Resonance Velocimetry using Physics-informed Neural Networks](https://mlsanity.com/paper/2107.07863/) (2021)
- [Forecasting Crude Oil Prices Using Reservoir Computing Models](https://mlsanity.com/paper/2306.03052/) (2023)
- [Machine learning for advancing low-temperature plasma modeling and simulation](https://mlsanity.com/paper/2307.00131/) (2023)
- [Adjoint Sensitivities of Chaotic Flows without Adjoint Solvers: A Data-Driven Approach](https://mlsanity.com/paper/2404.12315/) (2024)
- [Enhancing Computational Efficiency in Multiscale Systems Using Deep Learning of Coordinates and Flow Maps](https://mlsanity.com/paper/2407.00011/) (2024)
- [Using Echo-State Networks to Reproduce Rare Events in Chaotic Systems](https://mlsanity.com/paper/2505.16208/) (2025)
- [Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning](https://mlsanity.com/paper/2505.24099/) (2025)
- [Modeling Biological Multifunctionality with Echo State Networks](https://mlsanity.com/paper/2510.23940/) (2025)
- [Quantum Reservoir Computing with Physics-Informed Correction for Reduced-Order PDE Forecasting](https://mlsanity.com/paper/2608.23119/) (2026)

## Abstract (arXiv)

We propose a physics-informed Echo State Network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional loss function during the training, which is based on the system's governing equations. The additional loss function penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system and a truncation of the Charney-DeVore system. Compared to the conventional ESNs, the physics-informed ESNs improve the predictability horizon by about two Lyapunov times. This approach is also shown to be robust with regard to noise. The proposed framework shows the potential of using machine learning combined with prior physical knowledge to improve the time-accurate prediction of chaotic dynamical systems.