Adjoint-based machine learning for active flow control

We develop neural-network active flow controllers using a deep learning partial differential equation augmentation method (DPM). The end-to-end sensitivities for optimization are computed using adjoints of the governing equations without restriction on the terms that may appear in the objective function. In one-dimensional Burgers' examples with analytic (manufactured) control functions, DPM-based control is comparably effective to standard supervised learning for in-sample solutions and more effective for out-of-sample solutions, i.e., with different analytic control functions. The influence of the optimization time interval and neutral-network width is analyzed, the results of which influence algorithm design and hyperparameter choice, balancing control efficacy with computational cost. We subsequently develop adjoint-based controllers for two flow scenarios. First, we compare the drag-reduction performance and optimization cost of adjoint-based controllers and deep reinforcement learning (DRL)-based controllers for two-dimensional, incompressible, confined flow over a cylinder at Re = 100, with control achieved by synthetic body forces along the cylinder boundary. The required model complexity for the DRL-based controller is 4229 times that required for the DPM-based controller. In these tests, the DPM-based controller is 4.85 times more effective and 63.2 times less computationally intensive to train than the DRL-based controller. Second, we test DPM-based control for compressible, unconfined flow over a cylinder and extrapolate the controller to out-of-sample Reynolds numbers. We also train a simplified, steady, offline controller based on the DPM control law. Both online (DPM) and offline (steady) controllers stabilize the vortex shedding with a 99% drag reduction, demonstrating the robustness of the learning approach. For out-of-sample flows (Re = {50, 200, 300, 400}), both the online and offline controllers successfully reduce drag and stabilize vortex shedding, indicating that the DPM-based approach results in a stable model. A key attractive feature is the flexibility of adjoint-based optimization, which permits optimization over arbitrarily defined control laws without the need to match a priori known functions.