In this paper, we develop a novel neural network model with mapping and piecewise optimization technology for several classes of the linear singularly perturbed initial value and boundary value differential equations with variable coefficients. First, the Legendre polynomials are selected as the activation function of the artificial neural network, the mapping technology is employed to transform the original uniform partition points and the piecewise optimization technology is used to improve the calculation accuracy. Then, the solution of the linear singularly perturbed differential equations is solved by using the extreme learning machine optimization algorithm. Finally, the numerical experiments show that the developed method can effectively improve the accuracy of the calculation.

中文翻译：

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基于映射和分段优化技术的几类奇摄动微分方程的勒让德神经网络方法

在本文中，我们针对具有可变系数的几类线性奇异摄动初始值和边界值微分方程，开发了一种具有映射和分段优化技术的新型神经网络模型。首先，选择勒让德多项式作为人工神经网络的激活函数，采用映射技术对原始均匀分配点进行变换，采用分段优化技术提高计算精度。然后，使用极限学习机优化算法求解线性奇异摄动微分方程的解。最后，数值实验表明，该方法可以有效提高计算精度。