mlsanityData to 14 Oct 2026

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

arXiv 1909.11759 ↗First posted 23 Sep 2019cs.LG cs.NA math.NAcited by 35

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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.

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

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Citing papers per year: 2021: 7; 2022: 6; 2023: 6; 2024: 6; 2025: 8; 2026: 2.

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  1. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  2. 2022Overview frequency principle/spectral bias in deep learningcited by 69
  3. 2021Towards Understanding the Condensation of Neural Networks at Initial Trainingcited by 32
  4. 2025Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systemscited by 31
  5. 2022On the Activation Function Dependence of the Spectral Bias of Neural Networkscited by 28
  6. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  7. 2021Reproducing Activation Function for Deep Learningcited by 19
  8. 2024Addressing Spectral Bias of Deep Neural Networks by Multi-Grade Deep Learningcited by 14
  9. 2022Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equationscited by 11
  10. 2021Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEscited by 10
  11. 2022A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiencycited by 7
  12. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
23 more
  1. 2023Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equationcited by 5
  2. 2024Frequency-adaptive Multi-scale Deep Neural Networkscited by 5
  3. 2025Spectral Bias in Variational Quantum Machine Learningcited by 5
  4. 2021An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Networkcited by 4
  5. 2025Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potentialcited by 4
  6. 2022Machine Learning in Heterogeneous Porous Materialscited by 3
  7. 2024Physics and geometry informed neural operator network with application to acoustic scatteringcited by 3
  8. 2022On the Compatibility between Neural Networks and Partial Differential Equations for Physics-informed Learningcited by 2
  9. 2025Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attentioncited by 2
  10. 2026Layer-wise Geometric Approximation Rates for Deep Networkscited by 2
  11. 2024Physics Encoded Blocks in Residual Neural Network Architectures for Digital Twin Modelscited by 1
  12. 2025On the accuracy of implicit neural representations for cardiovascular anatomies and hemodynamic fieldscited by 1
  13. 2025Dynamic Reconstruction of Ultrasound-Derived Flow Fields With Physics-Informed Neural Fieldscited by 1
  14. 2021Parallel frequency function-deep neural network for efficient complex broadband signal approximationcited by 0
  15. 2023Phase-shifted Adversarial Trainingcited by 0
  16. 2023Deep Neural-network Prior for Orbit Recovery from Method of Momentscited by 0
  17. 2023Zero Coordinate Shift: Whetted Automatic Differentiation for Physics-informed Operator Learningcited by 0
  18. 2023Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problemscited by 0
  19. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  20. 2024Two-scale Neural Networks for Partial Differential Equations with Small Parameterscited by 0
  21. 2025Frequency-adaptive tensor neural networks for high-dimensional multi-scale problemscited by 0
  22. 2025Towards High-Fidelity Gaussian Splatting with Queried-Convolution Neural Networkscited by 0
  23. 2026Operator-informed initialization for Fourier features physics-informed neural networkscited by 0