Keywords: neural network, counter-propagation, genetic algorithm, structural optimization.

The main objective of this research is to improve the efficiency of the Counter- Propagation Neural net response in structural analysis and optimization. To achieve this, a modification has been made on the learning coefficients, which resulted in a higher performance. The net is trained by two different procedures, random and genetic generation of training pairs. To examine the efficiency of the net, different examples has been investigated. The results of genetic trained Counter-Propagation net and the random trained one are compared with the exact solution.. The purpose of using Genetic Algorithms (GAs) is mainly to investigate its efficiency in the net response.

Counter-Propagation neural network is a combination of two well-known algorithms: the self-organizing map of Kohonen and the Grossberg outstars. In the process of training, the weight matrices are computed internally. To define criteria for proximity of weight matrices in the Kohonen layer and input vector, parameter is defined as below [1,2]:

For the learning coefficient, Hecht-Neilsen suggests a number in the range . Adeli and Park[3] defines the learning coefficient, , as a function of iteration number in the following form:

parameters and are defined as the Kohonen and the Grossberg layers learning coefficients respectively, and is the number of iterations. Choosing small values for these coefficients results in a very slow training process. When net convergence is reached, a weight vector does not change in the next iterations. On the contrary, choosing large values for and causes a rapid training process, but the weight vector starts to fluctuate and proper convergence is not guaranteed. Hence, with respect to the relation between rate of training and stability of the weight vectors, it is necessary to have large values for the learning coefficients in order to have a fast learning rate at the beginning and decrease it as the training process is accomplished. For this purpose the following formula has been suggested and implemented in the net training algorithm. Improvements have been shown by example.

a,b : Learning coefficients; N: Number of iterations

Genetic Algorithms are computationally simple, but powerful in their search for improvement, and they are not limited by restrictive assumptions about the search space, such as continuity or existence of derivatives. Genetic Algorithms are search procedures based on the mechanics of natural genetic and natural selection. They combine the concept of the artificial of the artificial survival of the fittest with genetic operators abstracted from nature to form a powerful search mechanism[4,5]. The main objective is error reduction of the Counter-Propagation Neural net (CPN) response by application of the Genetic Algorithms. Genetic operators are applied so that the net is improved and error on the output units is reduced. By error minimizing of the net response and generating proper genetic training pairs, the overall performance of the net as compared with random training pairs is improved. The optimim design of a three spans girder with uniform load a cocentrated load at each mid span was considered. The results show that modifying parametes a and b, the learing coefficients of the Kohonen and the Grossberg layers will improve the net efficiency. To investigate the effects of other alternatives on the CPN neural net response, GAs are used in the training process. Based on the minimization of the error function and application of the genetic operators, Reproduction, Crossover, and Mutation proper net response is achieved. Stress analysis of plane and space trusses is considered, the overall performance of the net has been improved.

1

L. Szewczyk and P. Hajela, "Neural Network Approximations in a Simulated Annealing Based Optima Structural Design", Structural Optimization, 4:90-98, 1992. doi:10.1007/BF01759922

2

A. Iranmanesh, and A. Kaveh, "Structural Optimization by Gradient-Based Neural Networks", Int. J. Numer. Meth. Engng. 46, 297-311, 1999. doi:10.1002/(SICI)1097-0207(19990920)46:2<297::AID-NME679>3.3.CO;2-3

3

H. Adeli and H. S. Park, "Counterpropagation Neural Networks in Structural Engineering", Journal of Structural Engineering, 121: 1205-1211, 1995. doi:10.1061/(ASCE)0733-9445(1995)121:8(1205)

4

S. J. Wu and P. T. Chow, "Integrated Discrete and Configuration Optimization of Trusses Using Genetic Algorithms", Computers & Structures, Vol. 55, No.4, 695-702, 1995. doi:10.1016/0045-7949(94)00426-4

5

H. Adeli, N.T. Cheng, "Integrated Genetic Algorithm for Optimization of Space Structures", J. Aerosp. Engrg., ASCE, Vol. 6, No. 4, 315-329, 1993. doi:10.1061/(ASCE)0893-1321(1993)6:4(315)

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