  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 DNNs often achieve convergence in the low frequency range first,
and a series of moderately-sized DNNs are constructed and 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 higher frequency
content over a specific range at the the speed of convergence as in the low
frequency range. As a result, the proposed PhaseDNN is able to convert high
frequency learning to low frequency one, allowing a uniform learning to
wideband functions. The PhaseDNN will then be applied to find the solution of
high frequency wave equations in inhomogeneous media through both differential
and integral equation formulations with least square residual loss functions.
Numerical results have demonstrated the capability of the PhaseDNN in learning
high frequency functions and oscillatory solutions of interior and exterior
Helmholtz equations.
