论文标题
基于傅立叶特征的深域分解方法
A deep domain decomposition method based on Fourier features
论文作者
论文摘要
在本文中，我们提出了一种基于傅立叶特征的深域分解方法（F-D3M），用于部分微分方程（PDES）。当前，基于深度神经网络的方法是为了解决PDE的积极开发的，但是它们的效率可能会因高频模式的问题而退化。在这种新的F-D3M策略中，对空间域进行了重叠域的分解，因此可以将高频模式降低到相对较低的频率模式。在每个局部子域中，构建了多傅立叶特征网络（MFFNET），其中有效的边界和接口处理应用于相应的损失函数。我们提出了F-D3M的一般数学框架，验证其准确性并通过数值实验证明其效率。
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.
