Learning ergodic averages in chaotic systems
This work addresses improving prediction accuracy for chaotic systems, which is incremental as it builds on existing echo state networks by adding physical knowledge.
The authors tackled predicting time averages in chaotic systems by proposing a physics-informed hybrid echo state network (hESN) that integrates an imperfect physical model, reducing relative error from 48% to 7% in a thermoacoustic system.
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.