Robust, data-efficient active flow control using embedded deep learning

A neural-network flow control model is developed by optimizing over the Navier--Stokes equations. The model's weights are optimized using an embedded deep learning (DL) method, which solves adjoints of the governing equations to provide the end-to-end sensitivities of model parameters needed for optimization. This algorithm is more robust and data-efficient compared to purely data-driven algorithms such as reinforcement learning. 

First, a 1D Burgers equation is controlled to exhibit purely convective behavior. The performance of the embedded DL control is more accurate compared to a supervised learning model trained using an analytical control function, the error of which accumulates during the evolution process. Out-of-sample tests for different objective behaviors are similarly accurate, which illustrates the robustness of the learned control model.

Second, the drag and laminar vortex shedding in incompressible flow over a cylinder are reduced  via controlled body forces near the cylinder boundary.  A significant drag reduction demonstrates the effectiveness of the embedded controller.   Sparse input data near the cylinder boundary and the downstream area makes the training of the controller more efficient compared to a dense flow field dataset. The learned control performs better over a range of testing Reynolds numbers than comparable reinforcement learning-trained models despite the fact that it uses orders of magnitude fewer model parameters. Therefore, the embedded DL method is a robust and data-efficient algorithm with promise for complicated applications such as turbulence control.