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

TL;DR

This paper introduces a multi-scale deep neural network approach for efficiently solving oscillatory Stokes flows in complex domains, demonstrating improved convergence and ability to learn highly oscillatory solutions compared to traditional DNNs.

Contribution

The paper develops a multi-scale DNN method that effectively captures high-frequency oscillations in Stokes flows, outperforming standard DNNs in convergence and accuracy.

Findings

Faster convergence of MscaleDNN over normal DNNs.

MscaleDNN successfully learns highly oscillatory solutions.

Consistent error decay across different loss functions.

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 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.