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

2019/09/23 by Wei Cai, Xiaoguang Li, Cai, Wei +3 · 5 citations
Engineering · Materials Science · Physics and Astronomy · #Electromagnetic Simulation and Numerical Methods #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Magnetic Properties and Applications #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1909.11759

openalex publication_date 2019/09/23 · openalex created_date 2019/12/26 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Understanding training and generalization in deep learning by Fourier\n analysis
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

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• Dynamic Reconstruction of Ultrasound-Derived Flow Fields With Physics-Informed Neural Fields
• Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention
• Frequency-adaptive tensor neural networks for high-dimensional multi-scale problems

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