# Physics-Informed Echo State Networks for Chaotic Systems Forecasting

## TL;DR

This paper introduces a physics-informed Echo State Network that incorporates physical laws into training, significantly enhancing the prediction horizon for chaotic systems like the Lorenz system without additional data.

## Contribution

The paper presents a novel physics-informed ESN framework that enforces physical constraints during training, improving chaotic system forecasting accuracy.

## Key findings

- Improved predictability horizon by about two Lyapunov times.
- Enhanced forecasting accuracy without extra training data.
- Demonstrated on Lorenz system with promising results.

## Abstract

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 of the ESNs, which penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system, where the physics-informed ESNs improve the predictability horizon by about two Lyapunov times as compared to conventional ESNs. 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.