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.

我们提出了一种物理信息化的回声状态网络（ESN），以预测混沌系统的演化。与传统的ESN相比，经过物理通知的ESN经过训练，可以解决监督学习任务，同时确保其预测不违反物理定律。这是通过在训练过程中引入附加的损失函数来实现的，该函数基于系统的控制方程式。附加损失功能可对非物理预测进行惩罚，而无需任何其他训练数据。这种方法在混乱的Lorenz系统和Charney-DeVore系统的截断上得到了证明。与传统的ESN相比，具有物理知识的ESN可以将可预测性范围提高约Lyapunov两倍。该方法在噪声方面也很可靠。