PSI - Issue 5

S. Sahnoun et al. / Procedia Structural Integrity 5 (2017) 1267–1274 A.Saifi et al / Structural Integrity Procedia 00 (2017) 000 – 000

1273

7

according to its potential p j

  j

y 

f p

(5)

j

4.2. The network of nuerons A network of neurons consists in relating several formal neurons as shown in the diagram below.

Fig 9: Neural network model

The set of related network neurons allows by algorithms the resolution of complex scientific problems, thanks to an adjustment of the weights of the connections in a learning phase. In our case we have used the maximum absolute contrast values as the input variable for network learning and the corresponding thickness as output variables. 4.3. Choosing Network parameters The choice of the parameters of the neural network is a primordial stage, it consists of fixing: The activation function : the choice of the neural model always includes that of the activation function. Different transfer functions that can be used as a neuron activation function. Based on the work (Funahashi (1989), Cybenko, G. (1989)), only one hidden layer with logistic activation function was found to be relevant for approximating any nonlinear function with the desired accuracy. The equation of a logistic transfer function illustrated in the figure is given by: The used learning algorithm : the conjugate gradient algorithms, in particular Trainscg is efficient for problems of a function approximation. Indeed, we chose this algorithm for learning our neural model. The number of the network layers: a single layer is sufficient for solving our nonlinear problem. The number of neurons : In the literature there are several methods for determining the number of neurons in the hidden layer. In our case, using the nntool function in matlab, we have calculated the performance by the quadratic mean of the mse (7) deviations. The use of 12 hidden neurons gave the best result. x 1 e y 1    (6)

n

(7)

i 0   

2

n mse 1

i d s

( 

)

i

4.4. Result of the modeling by the chosen neural network We have used 70% of the data for training, 15% for validation and 15% for testing. And we have plotted in the