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

Adjoint methods offer a computationally cheap and accurate way to calculate the sensitivity of a quantity of interest with respect to all the system parameters. However, adjoint methods require the implementation of an adjoint solver, which can be cumbersome. In this work, we infer the adjoint solution from data via data-driven stability analysis with reservoir computing. First, we derive the adjoint of a reservoir computer, and compute the sensitivity of the acoustic energy of a prototypical thermoacoustic system with respect to its design parameters. Second, we improve generalizability and robustness by embedding the physical knowledge about the nonlinearity and the time-delayed nature of the thermoacoustic dynamics. Third, we employ the data-driven sensitivity provided by the adjoint of the trained network within a parameter optimization framework to minimise the acoustic energy. This work opens possibilities for data-driven gradient-based design optimization.