PSI - Issue 47

Zoltán Bézi et al. / Procedia Structural Integrity 47 (2023) 646–653 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Both training methods can be applied to static networks, where there is no feedback or delay, and to dynamic networks, although the incremental mode is more often used for dynamic networks such as adaptive filters. There are several models for solving different problems in neural networks, such as the perceptron model, the linear network or the backpropagation method for static networks (Caudill et al. (1992), Li et al. (2012)). The training process consists of four main steps: building the training dataset, creating the network, training the network and applying the network to new input parameters. There are several variants of the algorithm we use for backpropagation (Nielsen (1992)), among which the gradient descent method has been used, which can be implemented in two different ways, incremental and batch mode, the essence of which has been presented earlier. A big drawback is that they are too slow when it comes to practical problems. There are already algorithms that converge ten to a hundred times faster than the algorithms discussed earlier. All of these methods work in batch mode. These faster algorithms fall into two categories. The first category uses heuristic techniques (Butta et. al. (2021)) and the second uses optimization techniques. These include Conjugate gradient algorithms, Line Search Routines, Quasi-Newton algorithms and Levenberg-Marquardt algorithm. Among the methods listed above, the Levenberg Marquardt algorithms seems to be the fastest method for training medium-sized (up to hundreds of weights) feedforward neural networks. As with the quasi-Newton methods, this algorithm approximates the second-order training speed using the Jacobian matrix without computing the Hessian matrix. One of the further developed versions of these method is the Bayesian regularization backpropagation, which is a network training function that updates the weight and bias values according to the Levenberg-Marquardt optimization. It minimizes the combination of squared errors and weights, and then identifies the appropriate combination to generate a well-generalized network. This algorithm generally works best when the network inputs and targets are chosen to fall approximately in the range [-1,1]. In this work this training method was used (MacKay (1992)). 2.2. ANN method structure for GTN parameter determination For the determination of the GTN parameters, the input data is provided by the finite element simulation results, for which small notched flat tensile (NT) specimens with different (1, 2, 4 mm) notches were used, illustrated in Figure 2.

Fig. 2. Dimensions of notched flat tensile specimens and the FEM models.

In this case, the input data set resulted in 90 GTN parameter sets, where the GTN parameters were varied according to Table 1. based on Latin Hypercube Sampling (LSH). The idea is to randomly create points in the n dimensional space defined by the domain of variables. In comparison to the Monte-Carlo simulation, the field is partitioned according to the number of samples, the probability of the parts of the field is the same, so a more uniform distribution can be achieved. The distribution was created with a Python program, which is based on the literature (Sarkar (2022)). The GTN parameters 1 q , 2 q and standard deviation were given fixed values, thus changing a total of 5 parameters in the optimization task.