SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3285-A3312, January 2020.
In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The PhaseDNN makes use of the fact that common deep neural networks (DNNs) often achieve convergence in the low frequency range first, and constructs a series of moderately sized DNNs trained for selected high frequency ranges. With the help of phase shifts in the frequency domain, each of the DNNs will be trained to approximate the function's specific high frequency range at the speed of learning for low frequency. As a result, the proposed PhaseDNN is able to convert high frequency learning to low frequency learning, allowing a uniform learning of wideband functions. The PhaseDNN is then applied to learn solutions of high frequency wave problems in inhomogeneous media through the least squares residual of either differential or integral equations. Numerical results have demonstrated the capability of the PhaseDNN as a meshless method in general domains in learning high frequency functions and oscillatory solutions of interior and exterior Helmholtz problems.

中文翻译：

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用于高频近似和波动问题的相移深度神经网络

SIAM科学计算杂志，第42卷，第5期，第A3285-A3312页，2020年1月。
在本文中，我们提出了一种相移深层神经网络（PhaseDNN），该网络在逼近高频函数和波动方程解时提供均匀的宽带收敛。PhaseDNN利用了常见的深度神经网络（DNN）通常首先在低频范围内实现收敛这一事实，并构造了一系列针对选定的高频范围进行训练的中等大小的DNN。借助频域中的相移，将训练每个DNN，使其以学习低频的速度近似函数的特定高频范围。结果，提出的PhaseDNN能够将高频学习转换为低频学习，从而实现宽带功能的统一学习。然后，通过微分方程或积分方程的最小二乘残差，将PhaseDNN应用于学习非均匀介质中的高频波问题的解决方案。数值结果证明了PhaseDNN作为无网格方法在一般领域中学习高频函数以及内部和外部亥姆霍兹问题的振动解的能力。