Keywords: neural networks, backpropagation, genetic algorithms, fitting, parameter estimation, nonlinear problems.

Solving different technical or scientific tasks leads to problems, described by a system of partial differential equations. Through the last few decades the so-called artificial intelligence or soft-computing methods were developed as alternatives to traditional solutions of problems which are difficult to be defined, described or resolved using traditional ones. Our research deals particularly with a neural networks [1,4] which can be simply described as methods that are able to take experience-based data into account, such as observations or a coherence of measurements.

The behaviour of a neural network is determined by a preceding training process. It consists of finding so-called synaptic weights which have influence on the response of a neural network, depending on the different components of an input signal. For our class of problems, a layered neural network is the most suitable solution. It is usually combined with popular backpropagation algorithm for training. This method is proved to perform relatively well for traditional tasks where neural networks are usually used, for example as in technological processes controlling. However, it seems to be unsatisfactory for solving more complicated technical tasks, mainly for its tendency to fall into local extremes.

Genetic algorithms [2] are modern optimization methods which are based on an analogy with the evolution processes of living creatures during aeons. The training of neural networks could be considered as an optimization process as well, because it can be seen as a minimization problem for an error function applied on all training examples together.

The synaptic weights of a neural network act as variables of the error function. The number of these unknown values is often bigger than one thousand and it can cause serious difficulties. This is why the main emphasis during the previous work on optimization algorithms was put on the ability to solve multidimensional tasks and identification of local extremes. The algorithm called SADE [3] is the result of this development. It is based on the traditional genetic algorithm scheme using the operations of mutation, crossover and selection, and it is extended by the so-called local mutation. The synaptic weights are the real values, therefore the desired algorithm should be able to operate on real domains. Hence the crossover operator of traditional genetic algorithms was replaced by the so called differential operator.

The effectivity of training of a three layered neural network was tested on various problems using both backpropagation method and SADE algorithms. One of these problems is to estimate the function value of the third of three equidistant points from the values of the first two points. The complexity of this problem depends on the form of unknown function. For example, the difficult problem is the case of a periodic function of the following type:

The obtained results indicate that the strategy based on genetic algorithms is faster and more accurate compared to the backpropagation method. Solution of a group of test functions indicates that the SADE method yields significantly lower error (by a few orders) after the same number of iterations.

Then we test the SADE training for solving much more complicated civil engineering problem. It is an estimation of parameters of a constitutive model for concrete called microplane model. One way to do this is to fit these values using an experimentator's own experience. As one of more up-to-date approaches to estimate material parameters the neural network simulation could be employed. The results of computations show that the neural network trained by the SADE algorithm has the ability to predict the microplane model parametres with a satisfying precise.

1

G.Yagawa and H.Okuda: Neural networks in computational mechanics, CIMNE, 1996

2

Z.Michalewicz: Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, 1992

3

O. Hrstka, A. Kucerová, M. Leps and J. Zeman: A competitive comparison of different types of evolutionary algorithms. In Proceedings of the Sixth International Conference of Artificial Intelligence to Civil and Structural Engineering, Civil-Comp Press, 2001. doi:10.4203/ccp.74.37

4

L. H. Tsoukalas and R. E. Uhri: Fuzzy and neural approaches in engineering, John Willey & Sons, 1997

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