## Formulation of the Problem
### Artificial Neural Networks  
Neural networks were applied in this study to evaluate the natural frequencies of the considered structure. The most popular neural networks [35] are Multi-Layer Perceptrons (MLPs) or Feed-Forward Neural Networks (FFNNs) and Radial Basis Function (RBF) networks. A basic structure of the feed-forward neural network consists of fundamental processing components, called _units_ (neurons), arranged in layers. Data flow through a neural network via connections between units: data start from the input layer and then proceed through hidden layers, and finally reach the output layer to return the processed data. A neural network with \(N\) layers consists of one input layer, \(N-2\) hidden layers, and one output layer. Its _architecture_ can be described by \(I-H^{1}-H^{2}-...-H^{N-2}-O\), where \(I\) is the number of inputs, \(H^{i}\) is the number of neurons in the \(i^{th}\) hidden layer, and \(O\) is the number of neurons in the output layer. Each neuron performs a linear transformation (by means of coefficients called _weights_ and _biases_, all gathered in vector \(\mathbf{w}\)) followed by a nonlinear (usually sigmoidal) transformation prior to transmitting the signal to the subsequent layer. The dimension of vector \(\mathbf{w}\) can be specified as \(I\times H^{1}+H^{1}\times H^{2}+\cdots+H^{N-3}\times H^{N-2}+H^{N-2}\times O+ \sum_{i}H^{i}+O\).  
ANN architecture is partially determined by the problem to be solved. The number of units in the hidden layers, as well as the number of hidden layers themselves, is usually determined by a trial-and-error procedure. The calibration of the ANN (_training_ or _learning_ procedure) is based on _P patterns_ concerning the vector of the selected FE model parameters \(\mathbf{\Lambda}\) (herein consisting of lamination angles) and the corresponding eigenfrequency vector \(\mathbf{f}\), which is obtained by means of FEM. The computation of the optimal vector of a network's free parameters \(\mathbf{w}^{*}\) can be formulated as follows:  
\[\mathbf{w}^{*}=\operatorname*{arg\,min}_{w}\left\{\mathrm{g}(\mathbf{w}) \right\}\,, \tag{9}\]  
where  
\[\mathrm{g}(\mathbf{w})=\frac{1}{2P}\sum_{i=1}^{P}\|\mathbf{f}^{i}-\mathbf{y}^ {i}\|=\frac{1}{2P}\sum_{i=1}^{P}\|\mathbf{f}^{i}-\mathrm{ANN}(\Lambda^{i}, \mathbf{w})\| \tag{10}\]  
and \(\mathbf{y}^{i}=\mathrm{ANN}(\Lambda^{i},\mathbf{w})\) is the approximation of vector \(\mathbf{f}^{i}\) given by the ANN (in what follows, this formula is simplified to \(\mathbf{y}^{i}=\mathrm{ANN}(\Lambda^{i})\) in order to focus not on the ANN itself but on the input-output relation). For classical MLP, the optimization problem described by Equation (10) is usually solved by using a _back-propagation_ algorithm designed especially for ANN training.  
In the method described in this paper, instead of a standard MLP (_shallow networks_), so-called _deep networks_ are applied. Training neural networks with more than three hidden layers is called _deep learning_[36]. Deep learning allows for the application of computational models composed of multiple processing layers. This, in turn, enables the networks to learn data representation with multiple levels of abstraction. Deep networks, in contrast to a standard one-hidden-layer MLP, have a great number of hidden neurons, usually in a few hidden layers. The number of free network parameters gathered in vector \(\mathbf{w}\) is, therefore, significantly higher than that in the case of MLP application.  
Deep learning techniques can be introduced to deal with very large data sets with thousands or even hundreds of thousands of patterns, which is the case in many optimization problems with a large number of variables.  
All ANN training and testing processes were performed in the Mathematica (V12.0, Wolfram Research Inc., Champaign, IL, USA) environment [37], and the completed ANNs were then transferred to Matlab.