A Tailored Physics-informed Neural Network Method for Solving Singularly Perturbed Differential Equations

Yiwen Pang, Ye Li, Sheng-Jun Huang
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Abstract

Physics-informed neural networks (PINNs) have recently been demonstrated to be effective for the numerical solution of differential equations, with the advantage of small real labelled data needed. However, the performance of PINN greatly depends on the differential equation. The solution of singularly perturbed differential equations (SPDEs) usually contains a boundary layer, which makes it difficult for PINN to approximate the solution of SPDEs. In this paper, we analyse the reasons for the failure of PINN in solving SPDE and provide a feasible solution by adding prior knowledge of the boundary layer to the neural network. The new method is called the tailored physics-informed neural network (TPINN) since the network is tailored to some particular properties of the problem. Numerical experiments show that our method can effectively improve both the training speed and accuracy of neural networks.
求解奇摄动微分方程的定制物理信息神经网络方法
物理信息神经网络(pinn)最近被证明对微分方程的数值解是有效的,具有所需的小实际标记数据的优势。然而,PINN的性能在很大程度上取决于微分方程。奇异摄动微分方程(SPDEs)的解通常包含一个边界层,这使得PINN很难逼近SPDEs的解。在本文中,我们分析了PINN在求解SPDE时失败的原因,并通过在神经网络中加入边界层的先验知识提供了一种可行的解决方案。这种新方法被称为定制物理信息神经网络(TPINN),因为该网络是针对问题的某些特定属性定制的。数值实验表明,该方法可以有效地提高神经网络的训练速度和精度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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