Article Details

Abstract

Multiscale function approximation stands as a fundamental topic in computational mathematics and signal processing, with broad applications in seismic wave inversion, electrocardiogram signal analysis, wireless communication modeling, and other scientific and engineering domains. However, standard Deep Neural Networks (DNNs) encounter severe challenges in multiscale problems due to the Spectral Bias phenomenon—where network training dynamics naturally prioritize convergence to low-frequency components, thereby limiting approximation accuracy for high-frequency modes. To overcome this bottleneck, this paper proposes a coupled variable scaling neural network architecture. The method exploits the intrinsic tendency of standard DNNs to converge preferentially to low frequencies, decomposing the target multiscale signal into multiple low-frequency sub-bands via time-domain scaling transformation and frequency-domain phase shift techniques. Parallel learning is conducted by moderately-sized subnetworks, followed by inverse transformation for signal reconstruction, thus converting the high-frequency approximation problem into a low-frequency learning problem. Numerical experiments demonstrate that this architecture significantly enhances convergence efficiency and approximation accuracy, exhibiting promising adaptability and computational efficacy in multiscale function approximation tasks.

Keywords

DNN; coupled neural network; Fourier transform; variable scaling method

References

1. [1] Shi Yongmin, Qi Jun, Zhang Chengxue, etc Exploration of Methods for Predicting Cracks Using Seismic Waveform Analysis Technology [J]. Petroleum Geophysical Exploration, 2005 (2): 128-130, 5
2. [2] Olivine O. Parkfield: A Failed Earthquake Prediction Experiment [J]. Knowledge Guide, 2014 (4): 38-41
3. [3] Qian Binjie, Tong Guangming Research progress on electrocardiogram prediction of atrial fibrillation occurrence [J]. Heart Journal, 2025, 37 (6): 719-724
4. [4] Yu Xiaomei, Gao Fei, Li Wenjuan, etc Pattern Recognition of Lidar Communication Signal Waveforms for Connected Vehicles [J]. Laser Journal, 2023, 44 (9): 166-171
5. [5] Li Xinxin, Liu Kai, Huang Haiyan Research on the Application of Multi line Function Method in Curve Fitting [J]. Journal of Metrology, 2018, 39 (5): 716-719
6. [6] Tianzhen, Yang Zhiqiang, Shi Zhen, etc A Curve Fitting Center Coordinate Algorithm and Accuracy Analysis [J]. Surveying Science, 2016, 41 (11): 31-34, 55
7. [7] Ma Honglei, Liu Chenglong, Song Tao, etc A new method for fitting the circular curve of the longitudinal section of existing railway lines [J]. Journal of Wuhan University (Information Science Edition), 2016, 41 (12): 1696-1700
8. [8] Gu Tianqi, Zhang Lei, Ji Shijun, etc Curve fitting method for closed discrete points [J]. Journal of Jilin University (Engineering Edition), 2015, 45 (2): 437-441
9. [9] Li Xuejia, Feng Hongqi, Mei Yu, etc Geospatial distance calculation algorithm based on cubic polynomial fitting trigonometric function [J]. Computer Measurement and Control, 2016, 24 (5): 199-201206
10. [10] Ye Yaodong Application of B-splines in Fitting the Settlement Curve of Shield Tunnels [J]. Journal of Underground Space and Engineering, 2016, 12 (2): 511-515
11. [11] Fan Gaohui, Liu Shanghe, Wei Ming, etc Research on mathematical model of corona current based on neural network curve fitting [J]. High Voltage Technology, 2015, 41 (3): 1034-1041
12. [12] Zhu Xunlong, Yuan Guoliang, Zhou Peng Research on Character Segmentation Method of Brush Calligraphy under BP Neural Network [J]. Computer Simulation, 2021, 38 (10): 497-501
13. [13] Xu Z，Zhang Y，Zhang Y. Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks [PP/OL].（2024-05-17）[2025-09.16].https://arxiv.org/pdf/1901.06523.
14. [14] Rahaman N，Arpit D，Baratin A，et al. On the Spectral Bias of Deep Neural Networks [C]//International Conference on Machine Learning，2019：5301-5310.
15. [15] Wang S，Yu X，Perdikaris P. When and Why PINNs Fail to Train: A Neural Tangent Kernel Perspective [J/OL].Journal of Computational Physics，2022，449：110768.https://doi.org/10.1016/j.jcp.2021.110768.
16. [16] Cai W，Li X G，Liu L Z. A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems [J].SIAM Journal on Scientific Computing，2020，42（5）：A3285-A3312.
17. [17] Cai W，Li X G，Liu L Z. PhaseDNN: A Parallel Phase Shift Deep Neural Network for Adaptive Wideband Learning [PP/OL].（2019-05-03）[2025-09-16].https://arxiv.org/abs/1905.01389.
18. [18] Ko S，Park S. VS-PINN: A Fast and Efficient Training of Physics-Informed Neural Networks Using Variable-Scaling Methods for Solving PDEs With Stiff Behavior [J/OL].Journal of Computational Physics，2025，529：113860.https://doi.org/10.1016/j.jcp.2025.113860.
19. [19] Zeng Renyuan Partial differential equation solving based on deep neural networks [D]. Hefei: University of Science and Technology of China, 2022
20. [20] Huangjinping, Zhang Zhengbing Exploration of the Relationship between Convolution Theorem and Fourier Transform Properties and Their Applications [J]. Journal of Changjiang University (Self Science Edition), 2016, 13 (19): 29-31, 41
21. [21] Yang Lijuan, Zhang Baihua, Ye Xuzhen Fast Fourier Transform (FFT) and Its Applications [J]. Optoelectronic Engineering, 2004 (S1): 1-3, 7
22. [22] Zhang Shujie, Zhang Weiwei, Xi Yunqi, etc Spectral analysis of electrocardiogram signals based on Fourier transform [J]. Information Systems Engineering, 2012 (1): 133-134

About the authors

Chen Yaoyu (1997-), male, Han ethnicity, from Ankang, Shaanxi, is currently pursuing a master's degree in machine learning.