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一种求解二阶常微分方程近似解的P-SVM方法
发表时间：2024-01-16 浏览量：1245 下载量：83
| 全部作者： | 姚翊飞，杨晓忠 |
| 作者单位： | 华北电力大学数理学院信息与计算研究所 |
| 摘 要： | 微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题，本文基于微分方程的回归方程观点与解法，应用统计回归方法求解二阶常微分方程，并给出基于中心支持向量机（proximal support vector machine，P-SVM）在常微分方程的初值和边值问题上的近似解求法。通过在目标优化函数中添加偏置项，构建P-SVM回归模型，从而避免大规模求解线性方程组，得到结构简洁的最优解表达式。模型通过最小化训练样本点的均方误差和，在保证精度的同时，有效提高了近似解的计算速度。此外，形式简洁固定的解析解表达式也便于在实际应用中进行定性分析和性质研究。数值试验结果验证了P-SVM方法是一种高效可行的常微分方程求解方法。 |
| 关 键 词： | 计算数学；常微分方程的数值解法；中心支持向量机（P-SVM）；二阶常微分方程；回归模型 |
| Title： | P-SVM method for solving approximate solutions of second order ordinary differential equations |
| Author： | YAO Yifei, YANG Xiaozhong |
| Organization： | Institute of Information and Computing, Mathematics and Physics Department, North China Electric Power University |
| Abstract： | The computational solution of differential equations is crucial in theories and application in computer engineering. Due to the high computational complexity and discrete form in traditional methods, the statistical regression method is introduced to solve the second-order ordinary differential equation based on the thought of regression equation. Futhermore, an approximate solution model based on proximal support vector machine (P-SVM) is proposed for the initial value and boundary value problems of ordinary differential equations. By introducing an offset term in the objective optimization function and construct the P-SVM regression model, we avoid to computing the large-scale solution of linear equations and obtain an optimal solution expression with a simple structure. Our model is optimized by minimizing the sum of the mean square errors in the training samples. The calculation speed of our approximate solution is effectively improved without losing accuracy. Besides, our analytical solution expression also facilitates qualitative analysis and property research in practical applications. The numerical experiments demonstrate that P-SVM method is a feasible and efficient method for solving ordinary differential equations. |
| Key words： | computational mathematics; numerical solution of ordinary differential equations; proximal support vector machine (P-SVM); second order ordinary differential equations; regression model |
| 发表期数： | 2023年12月第4期 |
| 引用格式： | 姚翊飞，杨晓忠. 一种求解二阶常微分方程近似解的P-SVM方法[J]. 中国科技论文在线精品论文，2023，16（4）：427-438. |
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