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High-Precision Phase-Shift Transferable Neural Networks for High-Frequency Function Approximation and PDE Solution

Xuyang Gao, Liang Chen, Minqiang Xu, Jing Niu

Abstract

Neural network based methods have emerged as a promising paradigm for scientific computing, yet they face critical bottlenecks in high frequency function approximation and partial differential equation (PDE) solving.

High-Precision Phase-Shift Transferable Neural Networks for High-Frequency Function Approximation and PDE Solution

Abstract

Neural network based methods have emerged as a promising paradigm for scientific computing, yet they face critical bottlenecks in high frequency function approximation and partial differential equation (PDE) solving.

Paper Structure

This paper contains 23 sections, 75 equations, 8 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Construction of a Transferable Neural Feature Space for low-frequency functions
  3. Phase-Shift Transferable Neural Networks for High-Frequency Approximation
  4. An improved filtering method
  5. A Parallel Phase-Shift Transferable Neural Network (PPTNN) for the 1D Case
  6. A Parallel Phase-Shift Transferable Neural Network for the 2D Case
  7. A Coupled Phase-Shift Transferable Neural Network (CPTNN)
  8. Solving Differential Equations with CPTNN
  9. Linear Differential Operator
  10. Nonlinear Differential Operator
  11. Numerical Results
  12. Filtering Accuracy
  13. Accuracy of approximation of functions with PhaseTNN
  14. Effects of the shape parameter and network capacity
  15. 1D function approximation
  16. ...and 8 more sections

Figures (8)

  • Figure 1: Relative $L_2$ error vs. $\gamma$ for $f_1$ ($a = 1, 30, 100$) with TransNet, PPTNN and CPTNN.
  • Figure 2: Convergence behavior for $f_1$ ($a=30$). (Left) $f_1$; (Middle) PPTNN relative $L_2$ error vs. hidden neurons per sub-network; (Right) CPTNN relative $L_2$ error vs. total hidden neurons.
  • Figure 3: Relative $L_2$ error vs. shape parameter $\gamma$ for variable coefficient equation \ref{['eq:variable coefficient']}
  • Figure 4: Numerical solution and absolute error obtained by CPTNN for equation \ref{['eq:variable coefficient']}.
  • Figure 5: Numerical solution and absolute error obtained by CPTNN for Helmholtz equation \ref{['eq:Helmholtz equation']}
  • ...and 3 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3