Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations
This work addresses a domain-specific challenge in numerical analysis for researchers and practitioners dealing with oscillatory integral equations, offering an incremental improvement over standard DNN methods.
The authors tackled the problem of solving oscillatory Fredholm integral equations using deep neural networks, which struggle with high-frequency components due to spectral bias, by proposing a multi-grade deep learning model that effectively extracts multiscale information and overcomes this issue, as demonstrated in numerical experiments.
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.