Data-driven computation of adjoint sensitivities without adjoint solvers: An application to thermoacoustics

Abstract: Adjoint methods have been the pillar of gradient-based optimization for decades. They enable the accurate computation of a gradient (sensitivity) of a quantity of interest with respect to all system's parameters in one calculation. When the gradient is embedded in an optimization routine, the quantity of interest can be optimized for the system to have the desired behaviour. Adjoint methods require the system's governing equations and their Jacobian. We propose a computational strategy to infer the adjoint sensitivities from data when the governing equations might be unknown (or partly unknown), and noise might be present. The key component of this strategy is an echo state network, which learns the dynamics of nonlinear regimes with varying parameters, and evolves dynamically via a hidden state. Although the framework is general, we focus on thermoacoustics governed by nonlinear and time-delayed systems. First, we show that a parameter-aware Echo State Network (ESN) infers the parameterized dynamics. Second, we derive the adjoint of the ESN to compute the sensitivity to parameters and initial conditions. Third, we propose the Thermoacoustic Echo State Network (T-ESN), which embeds the physical knowledge in the network architecture. Fourth, we apply the framework to a variety of nonlinear thermoacoustic regimes of a prototypical system. We show that the T-ESN accurately infers the correct adjoint sensitivities of the time-averaged acoustic energy with respect to the flame parameters. The results are robust to noisy data, from periodic, through quasiperiodic, to chaotic regimes. A single network predicts the nonlinear bifurcations on unseen scenarios, and so the inferred adjoint sensitivities are employed to suppress an instability via steepest descent. This work opens new possibilities for gradient-based data-driven design optimization.