Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations

Jie Jiang, Yuesheng Xu

TL;DR

This work addresses solving oscillatory Fredholm integral equations of the second kind with neural methods, where the solution exhibits high-frequency components due to an oscillatory kernel. It introduces a discrete oscillatory quadrature $\mathcal{K}_{p_{κ}}$ with $p_{κ}=\lceil γ κ^{β}\rceil$ and analyzes the resulting operator $\mathbf{M}_{κ}$, proving invertibility on a co-countable set and providing an error bound that couples the training loss with quadrature error. To overcome neural spectral bias toward low frequencies, the paper proposes a multi-grade deep learning (MGDL) framework that stacks learning in multiple grades, extracting multiscale information and achieving superior accuracy over single-grade neural networks and classical collocation methods. Theoretical results bound the DNN approximation error in terms of the residual and quadrature error, and extensive numerical experiments demonstrate MGDL’s ability to recover high-frequency behavior and outperform existing baselines in oscillatory, multiscale settings. The approach offers a practically viable path for high-frequency, oscillatory integral equations in physics and engineering, with implications for quadrature design and spectral-bias mitigation in neural solvers.

Abstract

We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the oscillation of the kernel. It was pointed out recently that standard DNNs favour low frequency functions, and as a result, they often produce poor approximation for functions containing high frequency components. We addressed this issue in this study. We first developed a numerical method for solving the equation with DNNs as an approximate solution by designing a numerical quadrature that tailors to computing oscillatory integrals involving DNNs. We proved that the error of the DNN approximate solution of the equation is bounded by the training loss and the quadrature error. We then proposed a multi-grade deep learning (MGDL) model to overcome the spectral bias issue of neural networks. Numerical experiments demonstrate that the MGDL model is effective in extracting multiscale information of the oscillatory solution and overcoming the spectral bias issue from which a standard DNN model suffers.

Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations

TL;DR

This work addresses solving oscillatory Fredholm integral equations of the second kind with neural methods, where the solution exhibits high-frequency components due to an oscillatory kernel. It introduces a discrete oscillatory quadrature with and analyzes the resulting operator , proving invertibility on a co-countable set and providing an error bound that couples the training loss with quadrature error. To overcome neural spectral bias toward low frequencies, the paper proposes a multi-grade deep learning (MGDL) framework that stacks learning in multiple grades, extracting multiscale information and achieving superior accuracy over single-grade neural networks and classical collocation methods. Theoretical results bound the DNN approximation error in terms of the residual and quadrature error, and extensive numerical experiments demonstrate MGDL’s ability to recover high-frequency behavior and outperform existing baselines in oscillatory, multiscale settings. The approach offers a practically viable path for high-frequency, oscillatory integral equations in physics and engineering, with implications for quadrature design and spectral-bias mitigation in neural solvers.

Abstract

We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the oscillation of the kernel. It was pointed out recently that standard DNNs favour low frequency functions, and as a result, they often produce poor approximation for functions containing high frequency components. We addressed this issue in this study. We first developed a numerical method for solving the equation with DNNs as an approximate solution by designing a numerical quadrature that tailors to computing oscillatory integrals involving DNNs. We proved that the error of the DNN approximate solution of the equation is bounded by the training loss and the quadrature error. We then proposed a multi-grade deep learning (MGDL) model to overcome the spectral bias issue of neural networks. Numerical experiments demonstrate that the MGDL model is effective in extracting multiscale information of the oscillatory solution and overcoming the spectral bias issue from which a standard DNN model suffers.
Paper Structure (8 sections, 22 theorems, 175 equations, 6 figures, 4 tables)

This paper contains 8 sections, 22 theorems, 175 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Oscillatory Fredholm Integral Equation of the Second Kind
  3. DNN Learning Model
  4. Numerical Quadrature of Oscillatory Integrals
  5. Analysis of Single-Grade Learning Model
  6. Multi-Grade Learning Model
  7. Numerical Experiments
  8. Conclusive Remarks

Key Result

Theorem 2.1

Let $y$ be the solution of equation fredholm_equation. If $K\in C^{[m]}(I^2)$ where $C^{[n]}(I^2):=\{L\in C^n(I^2):max_{p,q\in \mathbb{Z}_{n+1}}\|L^{(p,q)}\|<\infty\}$ is independent of $\kappa$ and $f\in H^m_{\kappa,0}(I)$, then $y\in H^m_{\kappa,0}(I)$.

Figures (6)

  • Figure 1: Values of $\|\mathbf{M}_{\kappa}^{-1}\|_{N_\kappa}$ as a function of $\kappa$ for different $(\lambda, \gamma, \beta, q)$.
  • Figure 2: The training loss and validation loss for SGL-1, SGL-2, SGL-3 and MGDL for the case $\kappa=350, N_\kappa=12\kappa+1$.
  • Figure 3: Absolute errors of the approximate solutions of SGL-3 and MGDL at $s_j:=-1+j/10000$, $j\in \mathbb{Z}_{20001}$ for the case $\kappa:=350, N_\kappa:=12\kappa+1$.
  • Figure 4: Grade components of MGDL for the case $\kappa=350, N_\kappa=12\kappa+1$.
  • Figure 5: Relative errors of the approximate solutions generated by SGL-3 and grades 1, 2, 3 of MGDL in the frequency domain for the case $\kappa=350, N_\kappa=12\kappa+1$.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 2.1
  • Theorem 2.1
  • lemma 4.1
  • proof
  • lemma 4.2
  • proof
  • lemma 4.3
  • proof
  • lemma 4.4
  • proof
  • ...and 30 more