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.

本研究探讨了深度神经网络（DNNs）在解决二阶Fredholm积分方程的振荡问题上的应用。我们发展了一种利用DNNs作为近似解的数值方法，通过设计适用于涉及DNNs的振荡积分的数值积分方法来解决该方程，所提出的多重层次深度学习（MGDL）模型有效地提取了振荡解的多尺度信息，并克服了标准DNN模型所面临的光谱偏差问题。

深度神经网络解振荡费德霍尔姆积分方程