PSI - Issue 59

Victor Aulin et al. / Procedia Structural Integrity 59 (2024) 444–451

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Victor Aulin et al. / Structural Integrity Procedia 00 (2019) 000 – 000

optimal value. Data preprocessing is an essential step in models building by the ANN methodas the incorrectly chosen scaling range for the output data into the activation function's value will make a significant impact on the ANN method 's ability to generalize after training. This capability of the ANN methodcan beidentified by combining the defects which were in the training samling. The application of the mathematical apparatus of the ANN method as a method of intelligent analysis of objectshas been justified by Data Mining research. When constructing anANN method model with a single hidden layer, the application of the Rosenblatt rule for network training is limited due to the need to know the correct values at the inputs and outputs of the hidden layers, which is impossible. One of the most popular methods for training a perceptron with hidden layers is the gradient descent method. This method involves adjusting each specific weight coefficient 1 LL w  in the opposite direction of the gradient (antigradient) of the error estimation function at each new approximation (iteration). The method necessitates the adjustment of specific weights for each iteration. In the case of transmitting signals from the last hidden layer to the output layer, the adjustment is carried out according to the formula: where – is the algorithm learning speed. To assess the accuracy of neural network classification after training, a target error function ε is used. It depends on the specific weight coefficients and is a function of them. The most common target error evaluation function is the quadratic error function (Hinton et al., 2006):   2 1 1 2 y N d r L L i y y      , under a limited set of input data conditions and to evaluate the quality of ANN method training, the goal is to minimize the error function. Then, the target error function (12) takes the form: min   . (13) The error backpropagation method is an iterative gradient algorithm that is a modification of the classical gradient descent method. In this method, during the training process, the specific weights of neurons in the neural network are adjusted based on the signals received from the previous layer and taking into account the errors during the processing of layers in the reverse direction, starting from the last layer (the backpropagation principle). On the other hand, the error values can be minimized by increasing the number of training iterations of the network. The main requirement for applying this algorithm is the choice of a differentiated activation function. The diagram of a multilayer perceptron is provided in Figure 3. As we can see, the input layer 0  l contains a number of neurons i x . In this layer, only signal transmission occurs without any mathematical operations from the entering the next layer, hence its designation is different. The first hidden layer is denoted as 1  l . Each subsequent hidden layer is denoted as 2... 1   l L . In the hidden layers, the number of neurons is equal to 1 1,..., i L p p   . The final layer L is the output one, and the number of neurons in it is equal to L p . , (12) where y N – is the number of neurons of the output layer; d r L y – is the obtained (real) output vector of the neural network. To reduce the search space for the dependency ( ) Y f S  L y – is the desirable output vector of the neural network; 1 1 LL LL w     w      , (11)