利用机器学习技术确定同伦分析解中的收敛控制参数

doi:10.3969/j.issn.1000-5641.2022.02.005

华东师范大学学报（自然科学版） ›› 2022, Vol. 2022 ›› Issue (2): 34-44.doi: 10.3969/j.issn.1000-5641.2022.02.005

利用机器学习技术确定同伦分析解中的收敛控制参数

周童晖¹, 柳银萍^2,^3,*()

1. 华东师范大学计算机科学与技术学院, 上海　200062
2. 华东师范大学数学科学学院, 上海　200241
3. 华东师范大学上海市核心数学与实践重点实验室, 上海　200241

收稿日期:2020-11-25 出版日期:2022-03-25 发布日期:2022-03-28
通讯作者: 柳银萍 E-mail:ypliu@cs.ecnu.edu.cn
基金资助:
国家自然科学基金(11871328); 上海市自然科学基金(19ZR1414000); 上海市核心数学与实践重点实验室基金(18dz2271000)

Determination of convergence control parameters in homotopy analysis solutions based on machine learning technique

Tonghui ZHOU¹, Yinping LIU^2,^3,*()

1. School of Computer Science and Technology, East China Normal University, Shanghai　200062, China
2. School of Mathematical Sciences, East China Normal University, Shanghai 200241, China
3. Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

Received:2020-11-25 Online:2022-03-25 Published:2022-03-28
Contact: Yinping LIU E-mail:ypliu@cs.ecnu.edu.cn

摘要/Abstract

摘要：

同伦分析方法是求解强非线性问题解析近似解的有效方法, 已被广泛应用于解决科学研究和工程技术中的一些重要问题. 相对于其他已有的解析近似方法, 同伦分析方法通过引入若干个辅助参数和辅助函数来控制级数解的收敛区域和收敛速度. 针对现有的同伦分析方法中收敛控制参数的选择问题, 采用了一种根据机器学习的参数选择算法, 首次将同伦分析方法和机器学习技术结合起来, 求解非线性数学物理方程收敛性更好的解析近似解. 通过将该算法应用到具体的实例中, 可以看出, 所获得的同伦分析解明显优于已有的同伦分析解, 同时, 该算法更具普适性和灵活性.

关键词: 同伦分析方法, 辅助函数, 控制参数, 机器学习

Abstract:

Homotopy analysis method is an effective method for constructing approximate analytical solutions to strongly nonlinear problems. The technique has been widely applied to solve important problems in scientific research and engineering technology. Compared with other existing techniques, this method leverages auxiliary parameters and functions to adjust and control the convergence region and convergence speed of approximate analytical solutions. In this paper, we present a parameter selection algorithm based on machine learning techniques to determine the optimal values of convergence control parameters for homotopy analysis solutions. This marks the first time that homotopy analysis method and machine learning techniques have been combined to obtain approximate analytical method with better convergence for strongly nonlinear mathematical and physical equations. By applying the method to several examples, we show that the convergence of solutions using the proposed method is better than those obtained from existing homotopy analysis methods. In addition, our algorithm is both more universal and flexible.

Key words: homotopy analysis method, auxiliary function, control parameter, machine learning

中图分类号:

O175.14

周童晖, 柳银萍. 利用机器学习技术确定同伦分析解中的收敛控制参数[J]. 华东师范大学学报（自然科学版）, 2022, 2022(2): 34-44.

Tonghui ZHOU, Yinping LIU. Determination of convergence control parameters in homotopy analysis solutions based on machine learning technique[J]. Journal of East China Normal University(Natural Science), 2022, 2022(2): 34-44.

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参考文献 19

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epoch	$h $	MSE
20	0.981439173	48.66282272338860
1000	0.172596678	1.28252530097961
10000	–0.363417238	0.00061609549448
20000	–0.586410224	0.00000214512647
50000	–0.768192887	0.00000000528823
100000	–0.764399529	0.00000000507483

$t $	精确解	文献[4]的近似解	本文的近似解
0.1	0.09967	0.09967	0.09967
0.3	0.29131	0.29131	0.29133
0.5	0.46212	0.46212	0.46214
0.7	0.60437	0.60451	0.60438
0.9	0.71630	0.71839	0.71630
1.0	0.76159	0.76790	0.76161

${\{ {C_0},{C_1}\} _{(a,b)}} $	$\displaystyle\int_{0}^{1} N\left[u_{4}\left(x, C_{0}, C_{1}\right)\right]^{2} {\rm{d}} x$	$\displaystyle\int_{0}^{\pi} N\left[u_{4}\left(x, C_{0}, C_{1}\right)\right]^{2} {\rm{d}} x$
${\{ - 0.560\;35,0.414\;09\} _{(0,1)}} $	0.0337727	0.0370883
${\{ - 0.526\;03,0.390\;04\} _{(0,\pi )}} $	0.0458026	0.0359005

$x $	数值解^[18]	u(x)的 $10 $ 阶近似解
$x $	数值解^[18]	文献[13]	本文
0.05	0.93519000	0.93519000	0.93519000
0.50	0.60699000	0.60699000	0.60699000
5.00	0.07880800	0.07881400	0.07880800
20.00	0.00578490	0.00591100	0.00580070
50.00	0.00063226	0.00072607	0.00062749

$m $	MSE
$m $	文献[13]	本文
5	4 × 10^–5	5 × 10^–5
10	2 × 10^–9	1 × 10^–9
15	3 × 10^–13	6 × 10^–14
20	3 × 10^–14	2 × 10^–16

利用机器学习技术确定同伦分析解中的收敛控制参数

Determination of convergence control parameters in homotopy analysis solutions based on machine learning technique

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$m $	文献[13]		本文
$m $	$u′(0) $	相对误差/%	$u′(0) $	相对误差/%
5	–1.5876978885	2.3 × 10^–2	–1.5877665520	1.9 × 10^–2
10	–1.5880699210	9.6 × 10^–5	–1.5880702662	4.8 × 10^–5
20	–1.5880710127	5.9 × 10^–7	–1.5880710217	5.6 × 10^–8
40	–1.5880710222	2.1 × 10^–8	–1.5880710226	1.1 × 10^–9
60	–1.5880710226	3.0 × 10^–9	–1.5880710226	1.4 × 10^–11
数值解^[19]	–1.588071022611375313