In this paper we present a Fourier feature based deep domain decomposition method (F-D3M) for partial differential equations (PDEs). Currently, deep neural network based methods are actively developed for solving PDEs, but their efficiency can degenerate for problems with high frequency modes. In this new F-D3M strategy, overlapping domain decomposition is conducted for the spatial domain, such that high frequency modes can be reduced to relatively low frequency ones. In each local subdomain, multi Fourier feature networks (MFFNets) are constructed, where efficient boundary and interface treatments are applied for the corresponding loss functions. We present a general mathematical framework of F-D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.

翻译：在本文中,我们提出了一个基于Fourier地貌特征的局部差分方程深域分解法(F-D3M),目前,正在积极开发基于深神经网络的方法,以解决PDEs,但是由于高频模式的问题,其效率可能会下降。在这一新F-D3M战略中,空间域系的域别分解重叠,从而可以将高频模式降低到相对低频模式。在每个地方分域中,都建立了多个多Fourier地貌网络(MF FRFFFFFMetets),对相应的损失功能采用了高效的边界和界面处理。我们提出了一个F-D3M的一般数学框架,验证其准确性,并以数字实验来证明其效率。