  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.
