  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.
