  We develop a versatile optimization method, which finds the design parameters
that minimize time-averaged acoustic cost functionals. The method is
gradient-free, model-informed, and data-driven with reservoir computing based
on echo state networks. First, we analyse the predictive capabilities of echo
state networks both in the short- and long-time prediction of the dynamics. We
find that both fully data-driven and model-informed architectures 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,
whilst 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. 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 non-intrusive
("hands-off") optimization of chaotic systems, in which the cost of generating
data, for example from high-fidelity simulations and experiments, is high.
