A Linearized Learning with Multiscale Deep Neural Network for Stationary Navier-Stokes Equations with Oscillatory Solutions

In this paper, we combine a linearized iterative method with multi-scale deep neural network to compute oscillatory flows for stationary Navior-Stokes equation in complex domains. The multiscale neural network converts the high frequency components in target to low frequency components before training, thus accelerating the convergence of training of neural network for all frequencies. To solve the stationary nonlinear Navier-Stokes equation, we introduce the idea of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term. Three forms of linearizations will be considered. First we will conduct a benchmark problem of the linearized schemes in comparison with schemes based directly on the nonlinear PDE. Then, a Navier Stokes flow with high frequency components in a 2-D domain with a hole are learned by the linearized multiscale deep multiscale neural network. The results show that multiscale deep neural network combining with the linearized schemes can be trained fast and accurately.