Gradient-free optimization of chaotic acoustics with reservoir computing

Gradient-based optimization of chaotic acoustics is challenging for a threefold reason: (i) first-order perturbations grow exponentially in time; (ii) the statistics of the solution may have a slow convergence; and (iii) the time-averaged acoustic energy may physically have discontinuous variations, which means that the gradient does not exist for some design parameters. We develop a versatile optimization method, which finds the design parameters that minimize time-averaged acoustic cost functionals, and overcomes the three aforementioned challenges. The method is gradient-free, model-informed, and data-driven with reservoir computing based on echo state networks. First, we analyze the predictive capabilities of echo state networks in thermoacoustics both in the short- and long-time prediction of the dynamics. We find that both fully data-driven and model-informed architectures are able to learn the chaotic acoustic dynamics, both time-accurately and statistically. Informing the training with a physical reduced-order model with one acoustic mode markedly improves the accuracy and robustness of the echo state networks, while keeping the computational cost low. Echo state networks offer accurate predictions of the long-time dynamics, which would be otherwise expensive by integrating the governing equations to evaluate the time-averaged quantity to optimize. Second, we couple echo state networks with a Bayesian technique to explore the design thermoacoustic parameter space. The computational method is minimally intrusive because it requires only the initialization of the physical and hyperparameter optimizers. Third, we find the set of flame parameters that minimize the time-averaged acoustic energy of chaotic oscillations, which are caused by the positive feedback with a heat source, such as a flame in gas turbines or rocket motors. These oscillations are known as thermoacoustic oscillations. The optimal set of flame parameters 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. This work opens up new possibilities for nonintrusive (“hands-off”) optimization of chaotic systems, in which the cost of generating data, for example, from high-fidelity simulations and experiments, is high.