Physics

Helmholtz equation

Wideband

Mathematics

Economic growth

Evolutionary biology

Biology

Algorithm

Optics

Residual

Mathematical analysis

Function (biology)

Series (stratigraphy)

Paleontology

Boundary value problem

Artificial intelligence

Composite material

Economics

Frequency domain

Materials science

Range (aeronautics)

Fourier series

Helmholtz free energy

Convergence (economics)

Computer science

Phase (matter)

Quantum mechanics

Applied mathematics

Artificial neural network

SIAM Journal on Scientific Computing

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.

A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems