## 4 Numerical Example
### FNE Method
#### 4.3.2 Design of NN
The learning process of the network has been verified by mean-square-error, \(MSE(V)\); standard error, \(ste(V)\); and relative errors \(ep_{i}\), \(ep\), and \(epV_{avr}\):  
\[MSE(V)=\frac{1}{V\cdot M}\sum_{p=1}^{V}\sum_{p=1}^{M}\Big{(}z_{i}^{(p)}-y_{i} ^{(p)}\Big{)}^{2}, \tag{11}\]  
\[st\varepsilon(V)=\sqrt{\frac{1}{V\cdot M}\sum_{p=1}^{V}\sum_{p=1}^{M}\Big{(}z _{i}^{(p)}-y_{i}^{(p)}\Big{)}^{2}}, \tag{12}\]  
\[ep_{i}=\Big{(}1-z_{i}^{(p)}\Big{/}y_{i}^{(p)}\Big{)}\cdot 100\%,\,|ep_{i}|=\Big{|}1 -z_{i}^{(p)}\Big{/}y_{i}^{(p)}\Big{|}\cdot 100\%,\,ep=\frac{1}{M}\!\!\sum_{i=1}^{M} \!\!|ep_{i}|, \tag{13}\]  
\[eV_{\text{avr}}=\frac{1}{V\cdot M}\sum_{p=1}^{V}\sum_{i=1}^{M}\!|ep_{i}|, \tag{14}\]  
where \(z_{i}^{(p)}\) and \(y_{i}^{(p)}\) are the target and neurally computed \(i\)-th outputs for \(p\)-th pattern; \(M\) is the number of outputs; and \(V=L\),\(T\),\(P\)--number of learning (\(L\)), testing (\(T\)), and all (\(P\)) patterns, respectively. In Table 3, the calculated values of selected errors for prepared networks' architectures are presented. The successive phases in signal processing in the MPL network are not so-called \"black box\". They can be described as a relatively simple mathematical expression. However, for most tasks, there is no such need [39]. When developing any approximating formula, the simplest form is usually aimed at. This allows for the easier use of the equation in various computing environments. When the formula is derived from the structure and parameters of the neural network, the following principle should be followed: small network \(\rightarrow\) simple  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Number of Hidden Neurons H**} & \multicolumn{3}{c}{_ep (\%)_} & \multicolumn{3}{c}{**max \(ep_{i}\) (\%)**} \\ \cline{2-5}  & **L** & **T** & **L** & **T** \\ \hline
2 & 7.34 & 8.06 & 48.45 & 64.21 \\ \hline
3 & 1.01 & 1.80 & 4.22 & 10.79 \\ \hline
4 & 0.93 & 1.09 & 4.09 & 7.10 \\ \hline
5 & 0.30 & 0.37 & 2.43 & 3.93 \\ \hline
10 & 0.07 & 0.11 & 0.55 & 1 approximation formula. At the same time, all rules regarding the correctness of building NN for a given task should be kept. Figure 9 shows a graph of the percentage error--average (a) and maximum (b) depending on the number of neurons in the hidden layer. The optimal number of hidden neurons, H = 5, was picked out by using the cross-validation procedure (see Reference [40]). Moreover, the linear regression coefficient, \(r(V)\), was computed for every set of pairs. Figure 10 shows the correlation of load multipliers calculated by using FEM and NN. This correlation is close to unity for both the teaching and testing patterns. The output from the network, that is dependence \(\mu_{NN}(P,EA,Z)\) (15), can be formulated explicitly, through the known values of the network parameters:  
\[\mu_{NN}(P,EA,Z)=S_{\mu}\cdot(\mathbf{w_{YH}}\cdot(\tanh(\mathbf{w_{HX}}\cdot \left[ \begin{array}{cc}\frac{P}{S_{P}}&\frac{EA}{S_{LA}}&\frac{Z}{S_{Z}}\\ \end{array} \right]^{T}+\mathbf{b_{H}}))+\mathbf{b_{Y}}, \tag{15}\]  
where \(\mathbf{w_{HX}}\) is the weight of input (X) and hidden (H) layers connections; \(\mathbf{w_{YH}}\) is the weight of output (Y) and H layers connections; \(\mathbf{b_{H}}\) and \(\mathbf{b_{Y}}\) are the biases of neurons of subsequent layers of the network. Network parameters (weight and biases) are most often calculated iteratively in the learning process. After a decomposition, the formula on \(\mu_{NN}\) (15) for the vertical displacement of the truss top node, which is \(w_{1}\) = 7.20 cm, can be represented, for example, as follows (16):  
\[