## 4 Wavelet Neural Network (WNN) Model Building  
A wavelet neural network is constructed based on a BP neural network model and wavelet transform theory. It uses a wavelet function instead of an activation function of the hidden layer. It is a new type of backpropagation neural network [26]. The wavelet neural network integrates the time series localization of wavelet transform and the self-learning ability of a neural network, fundamentally solving the problem of localized minima and accelerating the convergence of the network. Its structure is generally divided into three layers: an input layer, a hidden layer, and an output layer. Each layer is passed through the activation function, and there are connection weights. The network structure is shown in Figure 7.  
The wavelet basis function of the hidden layer of the wavelet neural network is selected as the Morlet mother wavelet basis function, and the mathematical expression is shown in Equation (7):  
\[y=cos\ (1.75x)e^{-x^{2}/2} \tag{7}\]  
An image of the function is shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Serial** & \multirow{2}{*}{**Parameter Name**} & **Serial** & \multirow{2}{*}{**Parameter Name**} & **Serial** & \multirow{2}{*}{**Parameter Name**} \\
**Number** & & **Number** & & **Number** & \\ \hline X\({}_{2}\) & Furnace top pressure & X\({}_{7}\) & Furnace top temperature & X\({}_{14}\) & Oxygen enrichment rate \\ X\({}_{3}\) & Differential pressure & X\({}_{8}\) & Hot air pressure & X\({}_{15}\) & Cold air flow \\ X\({}_{4}\) & Gas utilization & X\({}_{9}\) & Coal injection volume & X\({}_{17}\) & Blast kinetic energy \\ X\({}_{5}\) & Oxygen enrichment & X\({}_{10}\) & Gas CO\({}_{2}\) volume fraction & X\({}_{21}\) & Furnace belly gas index \\ X\({}_{6}\) & Wind temperature & X\({}_{11}\) & Gas CO volume fraction & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Table of prediction model input parameters. Figure 7: Wavelet neural network structure. The formula for calculating the output of the implicit layer is shown in Equation (8):  
\[h(j)=h(\frac{\sum_{i=1}^{k}\omega_{ij}x_{i}-b_{j}}{a_{j}}),\ j=1,2,\ldots l \tag{8}\]  
where: \(x_{i}\) is the input value of the wavelet neural network; \(y\) is the predicted output value of the wavelet neural network; \(h(j)\) is the output value of the \(j\)th node in the hidden layer; \(h\) is the wavelet basis function; \(\omega_{ij}\) is the weight of the input layer to the hidden layer; \(a_{j}\) is the scaling factor of the wavelet basis function; and \(b_{j}\) is the translation factor of the wavelet basis function. The formula for the output layer is shown in Equation (9):  
\[y(k)=\sum_{i=1}^{l}\omega_{jk}h(i),\ k=1,2,\ldots,m \tag{9}\]  
where: \(\omega_{jk}\) is the weight from the hidden layer to the output layer; \(l\) and \(m\) are the number of nodes in the hidden layer and the output layer, respectively. A flow chart of the wavelet neural network prediction steps is shown in Figure 9.  
The prediction model for blast furnace permeability, along with the coupling mechanism analysis, enables real-time forecasting of blast furnace permeability. This ensures the continuous operation of the blast furnace and helps prevent any abnormal conditions from occurring. To keep the blast furnace running smoothly, it is necessary to optimize the charge quality and to maintain a normal furnace shape, an active furnace cylinder, adaptable gas distribution, a stable furnace temperature, reasonable slag composition, and good air permeability, etc., among which blast furnace air permeability is especially important. A data-driven blast furnace air permeability prediction model is constructed with 14 parameter variables, including the top pressure, blast kinetic energy, coal injection, differential pressure, cold air flow, belly gas index, gas utilization, hot air pressure, air temperature, top temperature, oxygen enrichment, oxygen enrichment rate, gas CO\({}_{2}\) volume fraction, and gas CO volume fraction.