Reduced-order deep learning for flow dynamics. The interplay between deep learning and model reduction

In this paper, we investigate neural networks applied to multiscale simulations and discuss a design of a novel deep neural network model reduction approach for multiscale problems. Due to the multiscale nature of the medium, the fine-grid resolution gives rise to a huge number of degrees of freedom. In practice, low-order models are derived to reduce the computational cost. In our paper, we use a non-local multicontinuum (NLMC) approach, which represents the solution on a coarse grid [18]. Using multi-layer learning techniques, we formulate and learn input-output maps constructed with NLMC on a coarse grid. We study the features of the coarse-grid solutions that neural networks capture via relating the input-output optimization to $l_1$ minimization of PDE solutions. In proposed multi-layer networks, we can learn the forward operators in a reduced way without computing them as in POD like approaches. We present soft thresholding operators as activation function, which our studies show to have some advantages. With these activation functions, the neural network identifies and selects important multiscale features which are crucial in modeling the underlying flow. Using trained neural network approximation of the input-output map, we construct a reduced-order model for the solution approximation. We use multi-layer networks for the time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. For a class of nonlinear problems, we suggest an efficient strategy. Numerical examples are presented to examine the performance of our method.