Gradient-free optimization of chaotic acoustics with echo state networks

Gradient-based optimisation of chaotic acoustics is challenging due to: (i) the exponential divergence of first-order perturbations (butterfly effect), (ii) the slow convergence rate of statistics and gradients, and (iii) the possibility that the gradient may not be defined for some design parameters. We propose a gradient-free methodology based on Bayesian optimization that overcomes all these issues. To bypass the high cost of time integration (due to slow convergence of statistics), we propose the use of echo state networks, which have been shown to predict chaotic systems with good performance. We analyze their short- and long-time predictive capabilities in thermoacoustics, finding that the network is capable of predicting the dynamics both time-accurately and statistically. Importantly, incorporating physical knowledge via a reduced-order model significantly improves the accuracy and robustness of the prediction. Finally, using the model-informed architecture, we find the set of heat source parameters that minimizes the time-averaged acoustic energy, a measure of the size of the chaotic acoustic oscillations. This optimal set is found with the same accuracy as brute-force grid search, but with a convergence rate that is more than one order of magnitude faster.