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.

翻译：我们研究了深度神经网络（DNNs）在第二类振荡型Fredholm积分方程数值求解中的应用。已知由于核函数的振荡特性，该方程的解呈现特定振荡行为。近期研究指出，标准DNNs偏向于低频函数，因此当处理包含高频分量的函数时，往往产生较差的逼近效果。本研究针对该问题展开探讨。我们首先设计了一种数值求积方法，通过定制可计算涉及DNNs的振荡积分，将DNNs作为近似解来求解方程。我们证明了方程DNN近似解的误差受训练损失和求积误差共同约束。进而提出多级深度学习（MGDL）模型以克服神经网络的谱偏差问题。数值实验表明，MGDL模型能有效提取振荡解的多尺度信息，并克服标准DNN模型所存在的谱偏差问题。