Multi-Scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains (Communications in Computational Physics, 2020)
In this paper, we study a multi-scale deep neural network (MscaleDNN)as a meshless numerical method for computing oscillatory Stokes flows in complexdomains. The MscaleDNN employs a multi-scale structure in the design of its DNNusing radial scalings to convert the approximation of high frequency components ofthe highly oscillatory Stokes solution to one of lower frequencies. The MscaleDNNsolution to the Stokes problem is obtained by minimizing a loss function in terms of $L^2$ norm of the residual of the Stokes equation. Three forms of loss functions are investigated based on vorticity-velocity-pressure, velocity-stress-pressure, and velocity-gradient of velocity-pressure formulations of the Stokes equation. We first conduct asystematic study of the MscaleDNN methods with various loss functions on the Kovasznay flow in comparison with normal fully connected DNNs. Then, Stokes flowswith highly oscillatory solutions in a 2-D domain with six randomly placed holes aresimulated by the MscaleDNN. The results show that MscaleDNN has faster convergence and consistent error decays in the simulation of Kovasznay flow for all threetested loss functions. More importantly, the MscaleDNN is capable of learning highlyoscillatory solutions when the normal DNNs fail to converge.