比例延迟微分方程的极限学习机算法

目的 针对比例延迟微分方程，提出一种基于极限学习机(ELM)算法的单隐藏层前馈神经网络训练方法，并将该方法推广到求解双比例延迟微分系统。方法 首先，构建一个单隐藏层前馈神经网络并随机生成输入权值和隐藏层偏置；然后，通过计算系数矩阵使其满足比例延迟微分方程及其初值条件，将其转化为最小二乘问题，利用摩尔-彭罗斯广义逆解出输出权值；最后，将输出权值代入构建的神经网络便可获得具有较高精度的比例延迟微分方程数值解。结果 通过数值实验与已有方法的结果进行比较，验证了该方法对处理比例延迟微分方程与双比例延迟微分系统的有效性，且随着选取的训练点和隐藏层节点数量增多，所得到的数值解精度和收敛速度也随之增加。结论 ELM算法对处理比例延迟微分方程以及双比例延迟微分系统具有较好的效果。

Extreme Learning Machine Algorithm for Pantograph Delay Differential Equations

Objective A single hidden layer feed-forward neural network training method based on extreme learningmachine ELM was proposed for pantograph delay differential equations and the method was extended to deal with thesystem of pantograph equations with two delays. Methods Firstly a feed-forward neural network with a single hiddenlayer was constructed and the input weights and hidden layer bias were randomly generated. Then by calculating thecoefficient matrix to satisfy the pantograph delay differential equation and its initial value conditions the equation wastransformed into a least squares problem and the output weight was obtained by using the Moor-Penrose generalizedinverse solution. Finally the numerical solution of the pantograph delay differential equation with high precision could beobtained by inputting the output weights into the constructed neural network. Results By comparing the results ofnumerical experiments with those of existing methods the effectiveness of the proposed method in dealing with pantographdelay differential equations and the system of pantograph equations with two delays was verified. With the increase in thenumber of selected training points and hidden layer nodes the accuracy and convergence rate of the numerical solutionswere also increased. Conclusion The ELM algorithm is effective in dealing with pantograph delay differential equationsand the system of pantograph equations with two delays.