  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.
