Unsupervised neural networks for solving Troesch’s problem

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

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.

Keywords: Troesch’s problem artificial neural network genetic algorithm hybrid methods

Received: 23 February 2013
Revised: 18 May 2013
Accepted manuscript online:

PACS:	89.20.-a	(Interdisciplinary applications of physics)
	89.20.Ff	(Computer science and technology)
	89.20.Kk	(Engineering)

Corresponding Authors: Muhammad Asif Zahoor Raja
E-mail: Muhammad.asif@ciit-attock.edu.pk, rasifzahoor@yahoo.com

Cite this article:

Muhammad Asif Zahoor Raja Unsupervised neural networks for solving Troesch’s problem 2014 Chin. Phys. B 23 018903

[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)

[1]	Adaptive genetic algorithm-based design of gamma-graphyne nanoribbon incorporating diamond-shaped segment with high thermoelectric conversion efficiency Jingyuan Lu(陆静远), Chunfeng Cui(崔春凤), Tao Ouyang(欧阳滔), Jin Li(李金), Chaoyu He(何朝宇), Chao Tang(唐超), and Jianxin Zhong(钟建新). Chin. Phys. B, 2023, 32(4): 048401.
[2]	Memristor's characteristics: From non-ideal to ideal Fan Sun(孙帆), Jing Su(粟静), Jie Li(李杰), Shukai Duan(段书凯), and Xiaofang Hu(胡小方). Chin. Phys. B, 2023, 32(2): 028401.
[3]	Characteristics of piecewise linear symmetric tri-stable stochastic resonance system and its application under different noises Gang Zhang(张刚), Yu-Jie Zeng(曾玉洁), and Zhong-Jun Jiang(蒋忠均). Chin. Phys. B, 2022, 31(8): 080502.
[4]	Pulse coding off-chip learning algorithm for memristive artificial neural network Ming-Jian Guo(郭明健), Shu-Kai Duan(段书凯), and Li-Dan Wang(王丽丹). Chin. Phys. B, 2022, 31(7): 078702.
[5]	Design optimization of broadband extreme ultraviolet polarizer in high-dimensional objective space Shang-Qi Kuang(匡尚奇), Bo-Chao Li(李博超), Yi Wang(王依), Xue-Peng Gong(龚学鹏), and Jing-Quan Lin(林景全). Chin. Phys. B, 2022, 31(7): 077802.
[6]	Parameter estimation of continuous variable quantum key distribution system via artificial neural networks Hao Luo(罗浩), Yi-Jun Wang(王一军), Wei Ye(叶炜), Hai Zhong(钟海), Yi-Yu Mao(毛宜钰), and Ying Guo(郭迎). Chin. Phys. B, 2022, 31(2): 020306.
[7]	A spintronic memristive circuit on the optimized RBF-MLP neural network Yuan Ge(葛源), Jie Li(李杰), Wenwu Jiang(蒋文武), Lidan Wang(王丽丹), and Shukai Duan(段书凯). Chin. Phys. B, 2022, 31(11): 110702.
[8]	A novel receiver-transmitter metasurface for a high-aperture-efficiency Fabry-Perot resonator antenna Peng Xie(谢鹏), Guangming Wang(王光明), Binfeng Zong(宗彬锋), and Xiaojun Zou(邹晓鋆). Chin. Phys. B, 2021, 30(8): 084103.
[9]	Refocusing and locating effect of fluorescence scattering field Jian-Gong Cui(崔建功), Ya-Xin Yu(余亚鑫), Xiao-Xia Chu(楚晓霞), Rong-Yu Zhao(赵荣宇), Min Zhu(祝敏), Fan Meng(孟凡), and Wen-Dong Zhang(张文栋). Chin. Phys. B, 2021, 30(12): 124210.
[10]	Artificial neural network potential for gold clusters Ling-Zhi Cao(曹凌志), Peng-Ju Wang(王鹏举), Lin-Wei Sai(赛琳伟), Jie Fu(付洁), and Xiang-Mei Duan(段香梅). Chin. Phys. B, 2020, 29(11): 117304.
[11]	Optimized dithering technique in frequency domain for high-quality three-dimensional depth data acquisition Ning Cai(蔡宁), Zhe-Bo Chen(陈浙泊), Xiang-Qun Cao(曹向群), Bin Lin(林斌). Chin. Phys. B, 2019, 28(8): 084202.
[12]	Multi-objective strategy to optimize dithering technique for high-quality three-dimensional shape measurement Ning Cai(蔡宁), Zhe-Bo Chen(陈浙泊), Xiang-Qun Cao(曹向群), Bin Lin(林斌). Chin. Phys. B, 2019, 28(10): 104210.
[13]	Broadband achromatic phase retarder based on metal-multilayer dielectric grating Na Li(李娜), Wei-Jin Kong(孔伟金), Feng Xia(夏峰), Mao-Jin Yun(云茂金). Chin. Phys. B, 2018, 27(5): 054202.
[14]	Electronic transport properties of lead nanowires Lishu Zhang(张力舒), Yi Zhou(周毅), Xinyue Dai(代新月), Zhenyang Zhao(赵珍阳), Hui Li(李辉). Chin. Phys. B, 2017, 26(7): 073102.
[15]	Optimization of multi-color laser waveform for high-order harmonic generation Cheng Jin(金成), C D Lin(林启东). Chin. Phys. B, 2016, 25(9): 094213.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Unsupervised neural networks for solving Troesch’s problem

Cite this article:

Online attention