| Abstract: | 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 $L^2$ 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 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 three
tested loss functions. More importantly, the MscaleDNN is capable of learning highly
oscillatory solutions when the normal DNNs fail to converge. |
