## 4 Numerical Example
### FNE Method
#### 4.3.2 Design of NN  
At the second stage of _FNE method_ to formulate explicit state functions at every marked equilibrium point (Figure 8) a feed-forward Multilayer Perceptron (MLP) with back-propagation of errors was used. MLP is the classic, most commonly used network structure in engineering issues. The neural network using the Neural Network Toolbox was designed [37]. It was formed by neurons arranged in layers with one direction of signal flow. In the later numerical analysis, the MLP with one hidden layer: X-H-Y, where X as the number of inputs, H as the number of neurons in hidden layer, and Y as the number of outputs were applied. After initial computations, for further analysis the group of networks with the MLP:3-H-1 architecture was accepted. The inputs to the networks were three parameters assumed in the task X = {_P_, _EA_, _Z_} as random variables (Table 2). The single output was the load multiplier, Y = {\(\mu\)}, in each of the 1-6 points of the equilibrium path. When using a MLP neural network equipped with sigmoidal activation functions, it is necessary to perform scaling or normalization of data provided as the network inputs and outputs. Lack of proper transformation causes serious disruptions in the learning process and inferior properties of the learned network. The expected output values should not take the upper or lower limit of the activation function. This range should not exceed [0; 1] and [\(-\)1; 1], respectively, for a sigmoid and a symmetrical (bipolar) sigmoid function of activation. For inputs data, there are no such strict limits; they can be scaled to the same values as the outputs. It is important that they are close to zero and have a small amplitude. Scaling is absolutely necessary when the data represent a different order of magnitude (see Table 2). The following simple formulae of scaling are proposed in this paper:  
\[S_{f,x}=\frac{\max\lvert\mathbf{x}\rvert}{1.05}, \tag{10}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Range of Values**} & \multicolumn{2}{c}{**Size Factor**} & \multicolumn{2}{c}{**After Scaling**} \\ \cline{2-5}  & **Min** & **Max** & **Min** & **Max** \\ \hline \(P\) (kN) & 8 & 12 & \(S_{P}=12.6\) & 0.6375 & 0.9524 \\ \hline _EA_ (kN) & 347004 & 518364 & \(S_{EA}=544280\) & 0.6375 & 0.9524 \\ \hline \(Z\) (m) & \(-0.1\) & 0.1 & \(S_{Z}=0.105\) & \(-0.9524\) & 0.9524 \\ \hline \(\mu\) & \multirow{2}{*}{dependent on the equilibrium point} & \multicolumn{2}{c}{\(S_{\mu}\)dependent on the equilibrium point} & \multicolumn{2}{c}{dependent on the equilibrium point} \\  & & & equilibrium point & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Features of task parameters. Figure 8: Equilibrium path \(\mu\) (q) and marked equilibrium points for which explicit neural formulas were developed. The example in the chart relates to the mean value of the random variables. where \(S_{fx}\) is the scaling factor of real value \(x\), and \(\mathbf{x}\) is the vector of real value \(x\). When determining the scale factor, the denominator was 1.05. This value was determined on the basis of our own numerical tests. The network usually achieves the best results when the denominator is in the range of 1.01 \(\div\) 1.10. Such a value causes that the scaled data does not exceed the range [\(-\)1, 1], i.e., the asymptotes limiting the bipolar sigmoid activation function. All the neural networks were trained by using the Levenberg-Marquardt (LM) learning algorithm. The LM method is an implementation of [PERSON]'s optimization strategy [38]. It is characterized by fast convergence, relatively easy calculations, and transparent implementation. The bipolar sigmoid activation function (\(\tanh(x)\)) in the hidden layers and the linear activation function in the output layer were assumed.