提出一种基于最小二乘支持向量机(LS-SVMs)的求解常微分方程近似解的改进方法.该方法首先通过离散计算域,将常微分方程转换为有约束条件的目标优化问题,然后利用径向基(RBF)核函数可导的性质,用带有导数形式的LS-SVM模型将此优化问题转化为LS-SVM回归问题,进而进行求解.最终得到的闭式近似解具有精度高、连续可微、结构简单且形式固定的特点.该方法适用于任意阶非刚性和奇异的线性常微分方程初值问题和边值问题,以及一阶非线性常微分方程问题.仿真结果验证了该方法具有良好的有效性.

In this paper we present a new method to solve ordinary differential equations (ODEs) by using Least Squares Support Vector Machines (LS-SVMs). We discretize the computational domain to make a transition from the ODEs to an optimization problem with constraint conditions, then transform the problem into a derivative formed LS-SVM regression  by using the differentiable RBF kernel and solve it. The high-accuracy  differentiable approximate solution with simple and fixed structure is   obtained in closed form. The method is applicable for solving any order non-stiff  and singular linear ODEs with initial or boundary conditions, and first order  nonlinear ODEs. Numerical simulation results demonstrate the efficiency of the method.