Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems

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.