收稿日期: 2020-11-25
网络出版日期: 2022-03-28
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
Received date: 2020-11-25
Online published: 2022-03-28
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 . DOI: 10.3969/j.issn.1000-5641.2022.02.005
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.
1 | VAN DYKE M. Perturbation Methods in Fluid Mechanics [M]. Stanford: Parabolic Press, 1975. |
2 | NAYFEH A H. Introduction to Perturbation Techniques [M]. New York: John Wiley & Sons, 1981. |
3 | NAYFEH A H. Perturbation Methods [M]. New York: John Wiley & Sons, 2008. |
4 | LIAO S J. Beyond Perturbation: Introduction to the Homotopy Analysis Method [M]. Boca Raton, Florida: CRC Press, 2004. |
5 | LIAO S J. Homotopy Analysis Method in Nonlinear Differential Equations [M]. Beijing: Higer Education Press, 2012. |
6 | LYAPUNOV A M. The general problem of the stability of motion. International Journal of Control, 1992, 55 (3): 531- 534. |
7 | ADOMIAN G. Nonlinear Stochastic Operator Equations [M]. Orlando: Academic Press, 1986. |
8 | 廖世俊. 求解非线性问题的同伦分析方法 [D]. 上海: 上海交通大学, 1992. |
9 | ODIBAT Z M. A study on the convergence of homotopy analysis method. Applied Mathematics and Computation, 2010, 217 (2): 782- 789. |
10 | YABUSHITA K, YAMASHITA M, TSUBOI K. An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. Journal of Physics A: Mathematical and Theoretical, 2007, 40 (29): 8403- 8416. |
11 | LIAO S J. Advances in the Homotopy Analysis Method [M]. Singapore: World Scientific, 2014. |
12 | VAN GORDER R A, VAJRAVELU K. On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: A general approach. Communications in Nonlinear Science and Numerical Simulation, 2009, 14 (12): 4078- 4089. |
13 | 赵银龙. 同伦分析方法之改进及其在非线性边值问题之应用 [D]. 上海: 上海交通大学, 2015. |
14 | ROUX N L, SCHMIDT M, BACH F. A stochastic gradient method with an exponential convergence rate for finite training sets [EB/OL].(2013-03-11)[2020-07-26]. https://arxiv.org/pdf/1202.6258.pdf. |
15 | KINGMA D P, BA J L . Adam: A method for stochastic optimization [EB/OL]. (2015-07-23)[2020-08-27]. https://arxiv.org/pdf/1412.6980v8.pdf. |
16 | FEYNMAN R P, METROPOLIS N, TELLER E. Equations of state of elements based on the generalized Fermi-Thomas theory. Physical Review, 1949, 75 (10): 1561- 1573. |
17 | COULSON C A, MARCH N H. Momenta in atoms using the Thomas-Fermi method. Proceedings of the Physical Society, 1950, 63 (4): 367- 374. |
18 | KOBAYASHI S, NAGAI S, UMEDA K. Accurate value of the initial slope of the ordinary TF function. Journal of the Physical Society of Japan, 1955, 10 (9): 759- 762. |
19 | FERNáNDEZ F M. Rational approximation to the Thomas-Fermi equations. Applied Mathematics and Computation, 2011, 217 (13): 6433- 6436. |
/
〈 |
|
〉 |