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

arXiv: https://arxiv.org/abs/1909.11759 · HTML: https://mlsanity.com/paper/1909.11759/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2019.

## Built on by (most cited first, 30 of 35)

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- [An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network](https://mlsanity.com/paper/2105.11675/) (2021)
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- [On the Compatibility between Neural Networks and Partial Differential Equations for Physics-informed Learning](https://mlsanity.com/paper/2212.00270/) (2022)
- [Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention](https://mlsanity.com/paper/2509.19331/) (2025)
- [Layer-wise Geometric Approximation Rates for Deep Networks](https://mlsanity.com/paper/2604.20219/) (2026)
- [Physics Encoded Blocks in Residual Neural Network Architectures for Digital Twin Models](https://mlsanity.com/paper/2411.11497/) (2024)
- [On the accuracy of implicit neural representations for cardiovascular anatomies and hemodynamic fields](https://mlsanity.com/paper/2510.20970/) (2025)
- [Dynamic Reconstruction of Ultrasound-Derived Flow Fields With Physics-Informed Neural Fields](https://mlsanity.com/paper/2511.01804/) (2025)
- [Parallel frequency function-deep neural network for efficient complex broadband signal approximation](https://mlsanity.com/paper/2106.10401/) (2021)
- [Phase-shifted Adversarial Training](https://mlsanity.com/paper/2301.04785/) (2023)
- [Deep Neural-network Prior for Orbit Recovery from Method of Moments](https://mlsanity.com/paper/2304.14604/) (2023)
- [Zero Coordinate Shift: Whetted Automatic Differentiation for Physics-informed Operator Learning](https://mlsanity.com/paper/2311.00860/) (2023)
- [Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems](https://mlsanity.com/paper/2311.03746/) (2023)

## Abstract (arXiv)

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.