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

doi:10.3969/j.issn.1000-5641.2022.02.005

Abstract

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

CLC Number:

O175.14

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.

Figures/Tables 8

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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