论文标题
多尺度的深神经网络（MSCALEDNN）方法，用于复杂域中的振荡性Stokes流动
Multi-scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains
论文作者
论文摘要
在本文中，我们研究了一个多尺度的深神经网络（MSCALEDNN），作为用于计算复杂域中振动性Stokes流量的无网数数值方法。 MSCALEDNN使用径向尺度在其DNN的设计中采用了多尺度结构，以将高度振荡性Stokes解决方案的高频分量近似转换为一个较低的频率之一。通过根据stokes方程的残差来最大程度地降低损失函数来获得stokes问题的MSCALEDNN解决方案。根据Stokes方程的速度 - 压力压力和速度级别的速度 - 压力压力和速度级别，研究了三种形式的损失函数。我们首先对MSCALEDNN方法进行了系统研究，与正常的完全连接的DNN相比，在Kovasznay流量上具有各种损失函数。然后，通过MSCALEDNN模拟了带有六个随机放置孔的2D结构域中的高度振荡溶液的Stokes流。结果表明，对于所有四个测试的损耗函数，MSCALEDNN具有更快的收敛性和一致的误差衰减。更重要的是，当正常DNN无法收敛时，MSCALEDNN能够学习高度振荡的溶液。
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.
