  In this paper, we study a multi-scale deep neural network (MscaleDNN) as a
meshless numerical method for computing oscillatory Stokes flows in complex
domains. The MscaleDNN employs a multi-scale structure in the design of its DNN
using radial scalings to convert the approximation of high frequency components
of the highly oscillatory Stokes solution to one of lower frequencies. The
MscaleDNN solution to the Stokes problem is obtained by minimizing a loss
function in terms of L2 normof 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 a systematic study of the
MscaleDNN methods with various loss functions on the Kovasznay flow in
comparison with normal fully connected DNNs. Then, Stokes flows with highly
oscillatory solutions in a 2-D domain with six randomly placed holes are
simulated by the MscaleDNN. The results show that MscaleDNN has faster
convergence and consistent error decays in the simulation of Kovasznay flow for
all four tested loss functions. More importantly, the MscaleDNN is capable of
learning highly oscillatory solutions when the normal DNNs fail to converge.
