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Adaptive self-organizing map using optimal control

The self-organizing map (SOM), which is a type of artificial neural network (ANN), was formulated as an optimal control problem. Its objective function is to minimize the mean quantization error, and the state equation is the weight updating equation of SOM. Based on the objective function and the s...

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Main Authors:	Alkawaz, Ali Najem, Kanesan, Jeevan, Badruddin, Irfan Anjum, Kamangar, Sarfaraz, Hussien, Mohamed, Baig, Maughal Ahmed Ali, Ahammad, N. Ameer
Format:	Article
Published:	MDPI 2023
Subjects:	TJ Mechanical engineering and machinery TK Electrical engineering. Electronics Nuclear engineering
Online Access:	http://eprints.um.edu.my/38272/
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spelling	my.um.eprints.382722023-06-23T02:25:55Z http://eprints.um.edu.my/38272/ Adaptive self-organizing map using optimal control Alkawaz, Ali Najem Kanesan, Jeevan Badruddin, Irfan Anjum Kamangar, Sarfaraz Hussien, Mohamed Baig, Maughal Ahmed Ali Ahammad, N. Ameer TJ Mechanical engineering and machinery TK Electrical engineering. Electronics Nuclear engineering The self-organizing map (SOM), which is a type of artificial neural network (ANN), was formulated as an optimal control problem. Its objective function is to minimize the mean quantization error, and the state equation is the weight updating equation of SOM. Based on the objective function and the state equations, the Hamiltonian equation based on Pontryagin's minimum principle (PMP) was formed. This study presents two models of SOM formulated as an optimal control problem. In the first model, called SOMOC1, the design is based on the state equation representing the weight updating equation of the best matching units of the SOM nodes in each iteration, whereas in the second model, called SOMOC2, it considers the weight updating equation of all the nodes in the SOM as the state updating equation. The learning rate is treated as the control variable. Based on the solution of the switching function, a bang-bang control was applied with a high and low learning rate. The proposed SOMOC2 model performs better than the SOMOC1 model and conventional SOM as it considers all the nodes in the Hamiltonian equation, and the switching function obtained from it is influenced by all the states, which provides one costate variable for each. The costate determines the marginal cost of violating the constraint by the state equations, and the switching function is influenced by this, hence producing a greater improvement in terms of the mean quantization error at the final iteration. It was found that the solution leads to an infinite order singular arc. The possible solutions for the suitable learning rates during the singular arc period are discussed in this study. MDPI 2023-04 Article PeerReviewed Alkawaz, Ali Najem and Kanesan, Jeevan and Badruddin, Irfan Anjum and Kamangar, Sarfaraz and Hussien, Mohamed and Baig, Maughal Ahmed Ali and Ahammad, N. Ameer (2023) Adaptive self-organizing map using optimal control. Mathematics, 11 (9). ISSN 2227-7390, DOI https://doi.org/10.3390/math11091995 <https://doi.org/10.3390/math11091995>. 10.3390/math11091995
institution	Universiti Malaya
building	UM Library
collection	Institutional Repository
continent	Asia
country	Malaysia
content_provider	Universiti Malaya
content_source	UM Research Repository
url_provider	http://eprints.um.edu.my/
topic	TJ Mechanical engineering and machinery TK Electrical engineering. Electronics Nuclear engineering
spellingShingle	TJ Mechanical engineering and machinery TK Electrical engineering. Electronics Nuclear engineering Alkawaz, Ali Najem Kanesan, Jeevan Badruddin, Irfan Anjum Kamangar, Sarfaraz Hussien, Mohamed Baig, Maughal Ahmed Ali Ahammad, N. Ameer Adaptive self-organizing map using optimal control
description	The self-organizing map (SOM), which is a type of artificial neural network (ANN), was formulated as an optimal control problem. Its objective function is to minimize the mean quantization error, and the state equation is the weight updating equation of SOM. Based on the objective function and the state equations, the Hamiltonian equation based on Pontryagin's minimum principle (PMP) was formed. This study presents two models of SOM formulated as an optimal control problem. In the first model, called SOMOC1, the design is based on the state equation representing the weight updating equation of the best matching units of the SOM nodes in each iteration, whereas in the second model, called SOMOC2, it considers the weight updating equation of all the nodes in the SOM as the state updating equation. The learning rate is treated as the control variable. Based on the solution of the switching function, a bang-bang control was applied with a high and low learning rate. The proposed SOMOC2 model performs better than the SOMOC1 model and conventional SOM as it considers all the nodes in the Hamiltonian equation, and the switching function obtained from it is influenced by all the states, which provides one costate variable for each. The costate determines the marginal cost of violating the constraint by the state equations, and the switching function is influenced by this, hence producing a greater improvement in terms of the mean quantization error at the final iteration. It was found that the solution leads to an infinite order singular arc. The possible solutions for the suitable learning rates during the singular arc period are discussed in this study.
format	Article
author	Alkawaz, Ali Najem Kanesan, Jeevan Badruddin, Irfan Anjum Kamangar, Sarfaraz Hussien, Mohamed Baig, Maughal Ahmed Ali Ahammad, N. Ameer
author_facet	Alkawaz, Ali Najem Kanesan, Jeevan Badruddin, Irfan Anjum Kamangar, Sarfaraz Hussien, Mohamed Baig, Maughal Ahmed Ali Ahammad, N. Ameer
author_sort	Alkawaz, Ali Najem
title	Adaptive self-organizing map using optimal control
title_short	Adaptive self-organizing map using optimal control
title_full	Adaptive self-organizing map using optimal control
title_fullStr	Adaptive self-organizing map using optimal control
title_full_unstemmed	Adaptive self-organizing map using optimal control
title_sort	adaptive self-organizing map using optimal control
publisher	MDPI
publishDate	2023
url	http://eprints.um.edu.my/38272/
_version_	1769842722330378240
score	13.211869

Adaptive self-organizing map using optimal control

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