Unsupervised neural networks for solving Troesch’s problem

doi:10.1088/1674-1056/23/1/018903

摘要/Abstract

摘要： In this study, stochastic computational intelligence techniques are presented for the solution of Troesch’s boundary value problem. The proposed stochastic solvers use the competency of a feed-forward artificial neural network for mathematical modeling of the problem in an unsupervised manner, whereas the learning of unknown parameters is made with local and global optimization methods as well as their combinations. Genetic algorithm (GA) and pattern search (PS) techniques are used as the global search methods and the interior point method (IPM) is used for an efficient local search. The combination of techniques like GA hybridized with IPM (GA-IPM) and PS hybridized with IPM (PS-IPM) are also applied to solve different forms of the equation. A comparison of the proposed results obtained from GA, PS, IPM, PS-IPM and GA-IPM has been made with the standard solutions including well known analytic techniques of the Adomian decomposition method, the variational iterational method and the homotopy perturbation method. The reliability and effectiveness of the proposed schemes, in term of accuracy and convergence, are evaluated from the results of statistical analysis based on sufficiently large independent runs.

关键词: Troesch’, s problem, artificial neural network, genetic algorithm, hybrid methods

Abstract: In this study, stochastic computational intelligence techniques are presented for the solution of Troesch’s boundary value problem. The proposed stochastic solvers use the competency of a feed-forward artificial neural network for mathematical modeling of the problem in an unsupervised manner, whereas the learning of unknown parameters is made with local and global optimization methods as well as their combinations. Genetic algorithm (GA) and pattern search (PS) techniques are used as the global search methods and the interior point method (IPM) is used for an efficient local search. The combination of techniques like GA hybridized with IPM (GA-IPM) and PS hybridized with IPM (PS-IPM) are also applied to solve different forms of the equation. A comparison of the proposed results obtained from GA, PS, IPM, PS-IPM and GA-IPM has been made with the standard solutions including well known analytic techniques of the Adomian decomposition method, the variational iterational method and the homotopy perturbation method. The reliability and effectiveness of the proposed schemes, in term of accuracy and convergence, are evaluated from the results of statistical analysis based on sufficiently large independent runs.

Key words: Troesch’s problem, artificial neural network, genetic algorithm, hybrid methods

中图分类号: (Interdisciplinary applications of physics)

89.20.-a

89.20.Ff (Computer science and technology) 89.20.Kk (Engineering)

Muhammad Asif Zahoor Raja. Unsupervised neural networks for solving Troesch’s problem[J]. Chin. Phys. B, 2014, 23(1): 18903-018903.

参考文献 56

[1]	Weibel E S 1958 The Plasma in Magentic Field (Landshoff R K M, Ed.) (Stanford: Stanford University Press)
[2]	Markin V S, Chernenko A A, Chizmadehev Y A and Chirkov Y G 1966 Aspects of the Theory of Gas Porous Electrodes (Bagotskii V S and Vasilev Y B, Ed.) (New York: Consultants Bureau) pp. 21–33
[3]	Godaspow D and Baker B S 1973 J. Electrochem. Soc. 120 1005
[4]	Troesch B A 1976 J. Comput. Phys. 21 279
[5]	Deeba E, Khuri S A and Xie S 2000 J. Comput. Phys. 159 125
[6]	Roberts S M and Shipman J S 1972 J. Comput. Phys. 10 232
[7]	Khuri S A 2003 Int. J. Comput. Math. 80 493
[8]	Chin R C Y and Krasny R 1983 SIAM J. Sci. Stat. Comput. 10 229
[9]	Momani S, Abuasad S and Odibat Z 2006 Appl. Math. Comput. 183 1351
[10]	Feng X, Mei L and He G 2007 Appl. Math. Comput. 189 500
[11]	Chang S H and Chang I L 2008 Appl. Math. Comput. 195 799
[12]	Chang S H 2010 J. Comput. Appl. Math. 234 3043
[13]	Chang S H 2010 Appl. Math. Comput. 216 3303
[14]	Khuri S A and Sayfy A 2011 Math. Comput. Model. 56 1907
[15]	Mirmoradi S H, Hosseinpour I, Ghanbarpour S and Barari A 2009 Applied Mathematical Sciences 3 1579
[16]	Mohamad A J 1979 J. Comput. Appl. Math. 5 171
[18]	Babolian E and Javadi S h 2004 Appl. Math. Comput. 153 253
[19]	Chin R C Y 1981 J. Comput. Appl. Math. 7 181
[20]	Parisi D R, Mariani M C and Laborde M A 2003 Chem. Eng. Processing 42 715
[21]	Khan J A, Raja M A Z and Qureshi I M 2011 Chin. Phys. Lett. 28 020206
[22]	Yazdi H S and Shahri R P 2010 Appl. Soft Comput. 10 267
[23]	Yazdan S, Hayati M and Moradian R 2009 Appl. Soft Comput. 9 20
[24]	Beidokhti R S and Malek A 2009 J. Franklin Institute 346 898
[25]	Khan J A, Raja M A Z and Qureshi I M 2011 Int. J. Phys. Sci. 6 7247
[26]	Khan J A, Raja M A Z and Qureshi I M 2011 Chin. Phys. Lett. 28 110205
[27]	Khan J A, Raja M A Z and Qureshi I M 2011 Ann. Math. Artif. Intell. 63 185
[28]	Raja M A Z, Khan J A, Ahmad S I and Qureshi I M 2013 Innovations in Intelligent Machines – 3, SCI 442 (Berlin Heidelberg: Springer-Verlag) pp. 103–117
[29]	Monterola C and Saloma C 2001 Opt. Express 9 72
[30]	El-Emam N N and Al-Rabeh R H 2011 Appl. Soft Comput. 11 3283
[31]	Raja M A Z, Ahmad S I and Samar R 2012 Neural Comput. Appl.
[32]	Raja M A Z, and Ahmad S I 2012 Neural Comput. Appl.
[33]	Raja M A Z submitted to Information Sciences Journal
[34]	Kumar M and Yadav N 2011 Comput. Math. Appl. 62 3796
[35]	Raja M A Z, Qureshi I M and Khan J A 2011 Int. J. Innov. Comput I 7 621
[36]	Raja M A Z, Khan J A and Qureshi I M 2010 Lecture Notes in Computer Science, Vol. 5990 (Berlin Heidelberg: Springer-Verlag) pp. 231–240
[37]	Raja M A Z, Khan J A and Qureshi I M 2010 GECCO (Companion) 2010 2023
[38]	Raja M A Z, Khan J A and Qureshi I M 2011 Math. Probl. Eng. 2011 76507501
[39]	Raja M A Z, Khan J A and Qureshi I M 2010 Ann. Math. Artif. Intell. 60 229
[40]	Aarts L P and Veer P V D 2001 Neural Process. Lett. 14 261
[41]	Khan J A and Raja M A Z 2013 Research Journal of Applied Sciences, Engineering and Technology 6 450
[42]	Karmarkar N 1984 Combinatorica 4 373
[43]	Wright S 1997 Primal-Dual Interior-Point Methods. (Philadelphia: SIAM)
[44]	Wright M H 2005 Bull. Amer. Math. Soc. 42 39
[45]	Hooke R and Jeeves T A 1961 Journal of the Association for Computing Machinery 8 212
[46]	Yu W C 1979 Positive Basis and a Class of Direct Search Techniques, (Zhongguo Kexue: Scientia Sinica) pp. 53–68
[47]	Dolan E D, Lewis R M and Torczon V J 2003 SIAM J. Optimiz. 14 567
[48]	Michael L R and Torczon V 1999 SIAM J. Optimiz. 9 1082
[49]	Michael L R and Torczon V 2000 SIAM J. Optimiz. 10 917
[50]	Zu Y X and Zhou J 2012 Chin. Phys. B 21 019501
[51]	Zu Y X and Zhou J 2011 Acta Phys. Sin. 60 079501 (in Chinese)
[52]	Zu Y X, Zhou J and Zeng CC 2010 Chin. Phys. B 19 119501
[53]	Wu P, He Y G and Fang G F 2013 Acta Phys. Sin. 62 020301 (in Chinese)
[54]	He R, Huang S X, Zhou C T and Jiang Z H 2012 Acta Phys. Sin. 61 049201 (in Chinese)
[55]	Zhou J, Liu Y A, Wu F, Zhang H G and Zu Y X 2011 Acta Phys. Sin. 60 090504 (in Chinese)
[56]	Wang J B and Lu J 2011 Acta Phys. Sin. 60 057304 (in Chinese)