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Adaptive Multi-Grade Deep Learning for Highly Oscillatory Fredholm Integral Equations of the Second Kind

Jie Jiang, Yuesheng Xu

TL;DR

The paper addresses solving highly oscillatory Fredholm integral equations of the second kind using an adaptive, multi-grade deep learning framework. It develops rigorous error analyses for both continuous and discrete MGDL models, showing stability and convergence under accurate quadrature and identifying the training error as the dominant source of approximation error. The Adaptive MGDL (AMGDL) algorithm automatically determines the necessary network grade from training performance, with theoretical guarantees linking training loss to true solution error. Numerical experiments with wavenumbers up to $\kappa=500$ and with singular solutions demonstrate substantial accuracy gains over single-grade models and establish AMGDL as a robust, scalable approach for multiscale oscillatory problems, including those with spectral bias challenges.

Abstract

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.

Adaptive Multi-Grade Deep Learning for Highly Oscillatory Fredholm Integral Equations of the Second Kind

TL;DR

The paper addresses solving highly oscillatory Fredholm integral equations of the second kind using an adaptive, multi-grade deep learning framework. It develops rigorous error analyses for both continuous and discrete MGDL models, showing stability and convergence under accurate quadrature and identifying the training error as the dominant source of approximation error. The Adaptive MGDL (AMGDL) algorithm automatically determines the necessary network grade from training performance, with theoretical guarantees linking training loss to true solution error. Numerical experiments with wavenumbers up to and with singular solutions demonstrate substantial accuracy gains over single-grade models and establish AMGDL as a robust, scalable approach for multiscale oscillatory problems, including those with spectral bias challenges.

Abstract

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.
Paper Structure (7 sections, 17 theorems, 169 equations, 6 figures, 10 tables, 1 algorithm)

This paper contains 7 sections, 17 theorems, 169 equations, 6 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Deep Learning Model
  3. Error Analysis of the Continuous MGDL Model
  4. Error Analysis of the Discrete MGDL Model
  5. Adaptive MGDL Algorithm
  6. Numerical Experiments
  7. Conclusion

Key Result

lemma thmcounterlemma

Let $\mathcal{B}_1$ and $\mathcal{B}_2$ be two given sets, and let $F : \mathcal{B}_1 \times \mathcal{B}_2 \to [0, +\infty)$ be a target function. If $(\theta_1^*, \theta_2^*) \in \mathcal{B}_1 \times \mathcal{B}_2$ is a local minimizer of the problem then $\theta_2^* \in \mathcal{B}_2$ is a local minimizer of the reduced problem

Figures (6)

  • Figure 1: Training and validation losses for AMGDL using equal-width networks.
  • Figure 2: Frequency-domain relative errors for AMGDL using equal-width networks: grades $1$–$7$.
  • Figure 3: Training and validation losses for AMGDL using varying-width networks.
  • Figure 4: Frequency-domain relative errors for AMGDL using varying-width networks: grades $1$–$8$.
  • Figure 5: Relative errors for AMGDL and SGDL models using equal-width networks across different singularity levels.
  • ...and 1 more figures

Theorems & Definitions (35)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • proposition thmcounterproposition
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 25 more