  In this paper, we propose a phase shift deep neural network (PhaseDNN) which
provides a wideband convergence in approximating a high dimensional function
during its training of the network. The PhaseDNN utilizes the fact that many
DNN achieves convergence in the low frequency range first, thus, a series of
moderately-sized of DNNs are constructed and trained in parallel for ranges of
higher frequencies. With the help of phase shifts in the frequency domain,
implemented through a simple phase factor multiplication on the training data,
each DNN in the series will be trained to approximate the target function's
higher frequency content over a specific range. Due to the phase shift, each
DNN achieves the speed of convergence as in the low frequency range. As a
result, the proposed PhaseDNN system is able to convert wideband frequency
learning to low frequency learning, thus allowing a uniform learning to
wideband high dimensional functions with frequency adaptive training. Numerical
results have demonstrated the capability of PhaseDNN in learning information of
a target function from low to high frequency uniformly.
