Adaptive Self-Organizing Map Using Optimal Control

Ali Najem Alkawaz, Jeevan Kanesan (), Irfan Anjum Badruddin (), Sarfaraz Kamangar, Mohamed Hussien, Maughal Ahmed Ali Baig and N. Ameer Ahammad
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Ali Najem Alkawaz: Department of Electrical Engineering, Faculty of Engineering, Universiti Malaya, Kuala Lumpur 50603, Malaysia
Jeevan Kanesan: Department of Electrical Engineering, Faculty of Engineering, Universiti Malaya, Kuala Lumpur 50603, Malaysia
Irfan Anjum Badruddin: Mechanical Engineering Department, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia
Sarfaraz Kamangar: Mechanical Engineering Department, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia
Mohamed Hussien: Department of Chemistry, Faculty of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
Maughal Ahmed Ali Baig: Department of Mechanical Engineering, CMR Technical Campus, Hyderabad 501401, Telangana, India
N. Ameer Ahammad: Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia

Mathematics, 2023, vol. 11, issue 9, 1-20

Abstract: 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.

Keywords: self-organizing map; artificial neural network; optimal control problem; Pontryagin’s minimum principle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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