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

TL;DR

This paper introduces an adaptive multi-grade deep learning method for solving highly oscillatory Fredholm integral equations of the second kind, demonstrating improved accuracy and robustness through rigorous analysis and numerical experiments.

Contribution

It proposes a novel adaptive MGDL algorithm that selects network grade based on training performance, enhancing solution accuracy for oscillatory integral equations.

Findings

The discrete MGDL model maintains convergence and stability with small quadrature error.

The adaptive algorithm effectively selects network grade based on training error.

Numerical experiments confirm high accuracy and robustness for challenging oscillatory problems.

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.