微分方程特征值问题的物理信息神经网络数值解法

Numerical solution of physically informed neural networks for eigenvalue problems of differential equations

摘要

针对微分方程特征值问题,提出一个改进的物理信息神经网络求解方法和两阶段训练法。该方法可以求解多个绝对值最小特征值、求解离初始值最近的特征值问题和重特征值问题等。通过一维、二维正方形区域以及L形区域中拉普拉斯算子特征值问题的数值算例表明本文方法比现有方法精度更高。

Abstract

For eigenvalue problems of differential equations, an improved physical information neural network solution and a two-stage training method are proposed. The new method can solve multiple minimum eigenvalues, the eigenvalue problem closest to the initial value and the multiple eigenvalue problem. Numerical examples of Laplace operator eigenvalue problems in 1D and 2D square regions as well as L-shaped regions show that the new method is more accurate than the existing methods.

关键词

物理信息神经网络 / 微分方程 / 特征值问题

Key words

physically informed neural networks / differential equations / eigenvalue problems

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基金资助

重庆市自然科学基金面上项目(CSTB2022NSCQ-MSX0610)