Data-driven computation of adjoint sensitivities without adjoint solvers: An application to thermoacoustics
arxiv(2024)
摘要
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 Jacobian, whose computation can
be cumbersome, and is problem dependent. We propose a computational strategy to
infer the adjoint sensitivities from data (observables), which bypasses the
need of the Jacobian of the physical system. 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 of time-averaged cost functionals. Third, we
propose the Thermoacoustic Echo State Network (T-ESN), which hard constrains
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.
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