In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they can be applied without much concerns on the form of the differential equations or the shape or dimension of the problem domain. When applied to problems with multi-frequency solutions, the performance and accuracy of neural network approximation methods are strongly affected by the contrast of the high- and low-frequency parts in the solutions. To address this issue, domain scaling and residual correction methods are proposed. The efficiency and accuracy of the proposed methods are demonstrated for multi-frequency model problems.

翻译：本文针对具有多频解的椭圆型偏微分方程发展了神经网络逼近方法。神经网络逼近方法相较于经典方法具有优势，可在无需过多关注微分方程形式或问题域形状与维度的前提下直接应用。当应用于多频解问题时，神经网络逼近方法的性能与精度会受到解中高低频分量对比度的显著影响。为解决该问题，本文提出了域缩放与残差修正方法。通过多频模型问题验证了所提方法的有效性与准确性。