We propose a physics-informed machine learning method to predict the time average of a chaotic attractor. The method is based on the hybrid echo state network (hESN). We assume that the system is ergodic, so the time average is equal to the ergodic average. Compared to conventional echo state networks (ESN) (purely data-driven), the hESN uses additional information from an incomplete, or imperfect, physical model. We evaluate the performance of the hESN and compare it to that of an ESN. This approach is demonstrated on a chaotic time-delayed thermoacoustic system, where the inclusion of a physical model significantly improves the accuracy of the prediction, reducing the relative error from 48% to 7%. This improvement is obtained at the low extra cost of solving two ordinary differential equations. This framework shows the potential of using machine learning techniques combined with prior physical knowledge to improve the prediction of time-averaged quantities in chaotic systems.

中文翻译：

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在混沌系统中学习遍历平均值

我们提出了一种基于物理的机器学习方法来预测混沌吸引子的时间平均值。该方法基于混合回声状态网络 (hESN)。我们假设系统是遍历的，所以时间平均值等于遍历平均值。与传统的回声状态网络 (ESN)（纯数据驱动）相比，hESN 使用来自不完整或不完善的物理模型的附加信息。我们评估 hESN 的性能并将其与 ESN 的性能进行比较。这种方法在混沌时滞热声系统上得到证明，其中包含物理模型显着提高了预测的准确性，将相对误差从 48% 降低到 7%。这种改进是通过求解两个常微分方程的低额外成本获得的。