  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.
