Backpropagation of neural network dynamical models applied to flow control

Backpropagation of neural network models (NNMs) is applied to control nonlinear dynamical systems using several different approaches. By leveraging open-loop data, we show the feasibility of building surrogate models with control inputs that are able to learn important features such as types of equilibria, limit cycles and chaos. Two novel approaches are presented and compared to gradient-based model predictive control (MPC): the neural network control (NNC), where an additional neural network is trained as a control law in a recurrent fashion using the nonlinear NNMs, and linear control design, enabled through linearization of the obtained NNMs. The latter is compared with dynamic mode decomposition with control (DMDc), which also relies on a data-driven linearized model. It is shown that the linearized NNMs better approximate the systems’ behavior near an equilibrium point than DMDc, particularly in cases where the data display highly nonlinear characteristics. The proposed control approaches are first tested on low-dimensional nonlinear systems presenting dynamical features such as stable and unstable limit cycles, besides chaos. Then, the NNC is applied to the nonlinear Kuramoto–Sivashinsky equation, exemplifying the control of a chaotic system with higher dimensionality. Finally, the proposed methodologies are tested on the compressible Navier–Stokes equations. In this case, the stabilization of a cylinder vortex shedding is sought using different actuation setups by taking measurements of the lift force with delay coordinates. Graphic abstract