2019

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

Cai, Wei, Li, Xiaoguang, Liu, Lizuo

Understand

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.

Built on

Brandt, Achi. Multi-level adaptive solutions to boundary-value problems. Mathematics of computation 31.138 (1977): 333-390
1977
Earlier work this paper cites.
Jiang BN, Povinelli LA. Least-squares finite element method for fluid dynamics. Computer Methods in Applied Mechanics and Engineering. 1990 Jul 1;81(1):13-37
1990
Earlier work this paper cites.
Daubechies, Ingrid. Ten lectures on wavelets. Vol. 61. Siam, 1992
1992
Earlier work this paper cites.
Bochev PB, Gunzburger MD. Finite element methods of least-squares type. SIAM review. 1998;40(4):789-837
1998
Earlier work this paper cites.

Similar

Goodfellow, Ian and Bengio, Yoshua and Courville, Aaron. Deep learning. MIT press(2016)
2016
Cited alongside, same era.
E, Weinan, and Bing Yu. ”The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.” Communications in Mathematics and Statistics 6.1 (2018): 1-12
2018
Cited alongside, same era.
Xu, Zhi-Qin John, Understanding training and generalization in deep learning by Fourier analysis, arXiv:1808.04295, November, 2018
Original
2018
Cited alongside, same era.
Cai, Wei, Xiaoguang Li, Lizuo Liu, PhaseDNN - A Parallel Phase Shift Deep Neural Network for Adaptive Wideband Learning, arxiv 1905.01389, May 3, 2019
2019
Cited alongside, same era.

Then

Luo, Tao and Ma, Zheng and Xu, Zhi-Qin John and Zhang, Yaoyu. Theory of the Frequency Principle for General Deep Neural Networks. arXiv preprint arXiv:1906.09235 (2019)
Original
2019
Closest in time.
Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378(2019): 686-707
2019
Closest in time.
Xu, Zhi-Qin John, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks. arXiv preprint arXiv:1901.06523 (2019)
Original
2019
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…