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In this paper, a multi-scale Fourier neural operator (MscaleFNO) is proposed to reduce the spectral bias of the FNO in learning the mapping between highly oscillatory functions, with application to the nonlinear mapping between the coefficient of the Helmholtz equation and its solution.
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W. Cai, X. Li, L. Liu, A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems
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Z. Liu, W. Cai, Z.-Q. J. Xu, Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains
2020
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Z.-Q. J. Xu, Y. Zhang, T. Luo, Y. Xiao, Z. Ma, Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks
2020
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C. R. Gin, D. E. Shea, S. L. Brunton, J. N. Kutz, DeepGreen: Deep Learning of Green’s Functions for Nonlinear Boundary Value Problems
2021
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L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
2021
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N. Boullé, C. J. Earls, A. Townsend, Data-driven Discovery of Green’s Functions with Human-understandable Deep Learning
2022
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N. B. Kovachki, Z.-Y. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. M. Stuart, A. Anandkumar, Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs
2023
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L. Zhang, W. Cai, Z.-Q. J. Xu, A Correction and Comments on ”Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains”
2023
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N. Boullé and A. Townsend, Chapter 3 - A mathematical guide to operator learning
2024
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X. Liu, B. Xu, S. Cao, and L. Zhang, Mitigating spectral bias for the multiscale operator learning
2024
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W. Cai. Deterministic, Stochastic, and Deep Learning Methods for Computational Electromagnetics
2025
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Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, Fourier Neural Operator for Parametric Partial Differential Equations
Cited in the paper.