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In this paper, we study a multi-scale deep neural network (MscaleDNN) as ameshless 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 componentsof the highly oscillatory Stokes solution to one of lower frequencies. TheMscaleDNN solution to the Stokes problem is obtained by minimizing a lossfunction in terms of L2 normof the residual of the Stokes equation. Three formsof loss functions are investigated based on vorticity-velocity-pressure,velocity-stress-pressure, and velocity-gradient of velocity-pressureformulations of the Stokes equation. We first conduct a systematic study of theMscaleDNN methods with various loss functions on the Kovasznay flow incomparison with normal fully connected DNNs. Then, Stokes flows with highlyoscillatory solutions in a 2-D domain with six randomly placed holes aresimulated by the MscaleDNN. The results show that MscaleDNN has fasterconvergence and consistent error decays in the simulation of Kovasznay flow forall four tested loss functions. More importantly, the MscaleDNN is capable oflearning highly oscillatory solutions when the normal DNNs fail to converge.

Multi-scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains