## Theoretical Framework  
### Network Training  
The Sum Square Error (SSE) is a performance index and it is used in order to evaluate the training of the network:  
\[\mathbf{E}=\frac{1}{2}\sum\
olimits_{p=1}^{N_{p}}\sum\
olimits_{\dot{o}=1}^{ N_{b}}e_{\dot{p}\dot{o}}^{2}=\frac{1}{2}\sum\
olimits_{p=1}^{N_{p}}\sum \
olimits_{\dot{o}=1}^{N_{o}}\left(t_{\dot{p}\dot{o}}-y_{\dot{p}\dot{o}}\right)^ {2}, \tag{4}\]  
where \(N_{p}\) is the number of patterns; \(N_{o}\) is the number of outputs; \(e_{\dot{p}\dot{o}}\) is the error value for the pattern \(p\) at the output \(\dot{o}\); \(t_{\dot{p}\dot{o}}\) is the target value for the pattern \(p\) at the output \(\dot{o}\); and \(y_{\dot{p}\dot{o}}\) is the output value of the network by the pattern \(p\) at the output \(\dot{o}\). The derivative of the error, as a function of the weights, defines the gradient, which gives information to find a minimum. The mathematical form is as follows:  
\[\mathbf{G}=\frac{\partial\mathbf{E}(\mathbf{w})}{\partial\mathbf{w}}, \tag{5}\]  
where \(\mathbf{G}\) is the gradient vector; and \(\mathbf{w}\) is the weight vector. The use of the gradient as principal parameter for reducing the error is known as the descendent gradient method. The update of weights to minimize the error by descendent gradient can be written as:  
\[\mathbf{w}+\Delta\mathbf{w}=\mathbf{w}-\alpha\mathbf{G}, \tag{6}\]  
where \(\alpha\) is the learning constant and \(\Delta\mathbf{w}\) the update for the weights. This method is stable but could be very slow with a lot of interactions due to inefficient values of \(\alpha\). The [PERSON] method allows for reducing the number of interactions using a square matrix of second-order partial derivatives. This matrix can be defined by Taylor series. According to [PERSON], it is possible to represent a function from an infinite sum of terms that are calculated by derivatives and powers in a single point. The Taylor polynomial of second degree can be written in its compact form as:  
\[\mathbf{T}(\mathbf{x})\approx\mathbf{T}(\mathbf{a})+\mathbf{G}(\mathbf{a})( \mathbf{x}-\mathbf{a})+1/2(\mathbf{x}-\mathbf{a})^{\mathrm{T}}\mathbf{H}( \mathbf{a})(\mathbf{x}-\mathbf{a}), \tag{7}\]  
where \(\mathbf{a}\) is the derivation point, \(\mathbf{G}(\mathbf{a})\) is the gradient vector and \(\mathbf{H}(\mathbf{a})\) is the Hessian matrix. Using the Taylor approximation, Equation (7), to estimate the next possible value of the error function, i.e., \(\mathbf{T}(\mathbf{x})=\mathbf{E}(\mathbf{w}+\Delta\mathbf{w})\), and considering the derivation point at \(\mathbf{a}=\mathbf{w}\), the expression would be:  
\[\mathbf{E}(\mathbf{w}+\Delta\mathbf{w})\approx\mathbf{E}(\mathbf{w})+ \mathbf{G}(\mathbf{w})\Delta\mathbf{w}+1/2\Delta\mathbf{w}^{\mathrm{T}} \mathbf{H}(\mathbf{w})\Delta\mathbf{w}. \tag{8}\]  
Assuming that the value of error is zero \(\mathbf{E}(\mathbf{w}+\Delta\mathbf{w})=0\) with the aim to close the error to a minimum, it is possible to reduce the expression with the derivative with respect to \(\Delta\mathbf{w}\):  
\[0=\mathbf{G}(\mathbf{w})+\mathbf{H}(\mathbf{w})\Delta\mathbf{w}, \tag{9}\]  
therefore:  
\[\Delta\mathbf{w}=-\mathbf{H}(\mathbf{w})^{-1}\mathbf{G}(\mathbf{w}), \tag{10}\]  
written in matrix form :  
\[\left[ \begin{array}{c}\Delta\mathbf{w}_{1}\\ \Delta\mathbf{w}_{2}\\ \vdots\\ \Delta\mathbf{w}_{N_{w}}\end{array} \right]=-\left[