## 2 Methods
### Adaptive Neuro-Fuzzy Inference System (ANFIS)  
The fuzzy neural network is a combination of neural networks and fuzzy logic, which offers the advantages of both linear and non-linear processes [18]. The neural network is composed of a set of connected artificial neurons using computational connection methods to process information [19]. The adaptive neural network (ANN) can understand the relationships between the data input and output when it has sufficient information [20]. The calculation principles and methods of ANN can be found in the literature [21]. Recently,neural networks and fuzzy logic systems have garnered significant attention in the field of intelligent control, as they can adjust inputs and outputs through a hybrid algorithm that combines back-propagation learning methods with the least squares method. The hybrid algorithm can also automatically generate if-then rules while allowing the fuzzy logic system to adapt membership function parameters. This enables the relevant FIS to detect and trace the given input and output data [22; 23]. The nonlinear mapping ability of neural networks allows for massive information storage, error tolerance, self-adaptive learning, and the ability to store incorrect information. These capabilities drive the ANFIS system toward self-adaptation, self-organization, and self-learning, making intelligence, self-adaptation, and optimization the primary development trend of ANFIS. The ANFIS model is composed of five layers, with each layer consisting of several nodes. Similar to neural networks, the inputs of each layer are obtained by the nodes from the previous layer in the ANFIS structure. Figure 1 describes an ANFIS structure. Figure 1 shows that the neural network contains \(m\) inputs (\(X_{1},X_{2},\ldots X_{m}\)), each one comprising \(n\) MFs; in addition, a layer with R fuzzy rules and an output layer contributed to the construction of the model. In the first layer, the number of nodes can be calculated by \(N=m\times n\). The number of nodes in other layers (layers 2-4) is related to the number of fuzzy rules (R) [24; 25]. Each layer in an ANFIS model is described as follows:  
The boxes in the figure mean that the relevant parameter can be adjusted adaptively at each node, and the circle is the opposite. The first layer is the fuzzy layer; input data are fuzzed in this layer and input data are converted to linguistic type \(A_{ij}\) using membership functions. The output of the first layer is as follows:  
\[O_{ij}^{1}=\mu_{ij}(X_{i}),i=1,2,\ldots,m,j=1,2,\ldots n \tag{4}\]  
where \(\mu_{ij}\) is the \(j\)th MF for input \(X_{i}\). Figure 1: The structure of the ANFIS model. The second layer is the product layer. The output of each node represents the excitation intensity of a fuzzy rule, and each node can be gained by multiplying the linguistic inputs calculated in the first layer:  
\[O_{k}^{2}=W_{k}=\mu_{1e_{1}}(X_{1})\mu_{2e_{2}}(X_{2})\ldots\mu_{me_{m}}(X_{m}) \tag{5}\]  
where \(k=1,2,\ldots,R\) and \(e_{1},e_{2},\ldots,e_{m}=1,2,\ldots n\). The third layer is the normalized layer: the ratio of the incentive intensity of the jth fuzzy rule to the sum of the incentive intensities of all rules:  
\[O_{k}^{3}=\overline{W}_{k}=\frac{W_{k}}{W_{1}+W_{2}+\ldots+W_{k}} \tag{6}\]  
The fourth layer is the defuzzification layer; each node of this layer is an adaptive node with node function, and the weighted output of each node depends on the if-then rules. The output of the fourth layer is as follows:  
\[O_{k}^{4}=\overline{W}_{k}f_{k} \tag{7}\]  
where \(f_{k}\) represents the output of \(k\)th fuzzy rules. The rules are expressed as follows:  
If (\(X_{1}\) is \(A_{1e_{1}}\)) and (\(X_{2}\) is \(A_{2e_{2}}\)) and \(\ldots\) and (\(X_{m}\) is \(A_{me_{m}}\)), then:  
\[f_{k}=\sum_{1=1}^{m}p_{ie_{1}}X_{i}+r_{k} \tag{8}\]  
where \(p_{ie_{i}}\) and \(r_{k}\) are the consequent parameters, \(e_{1},e_{2},\ldots,e_{m}=1,2,\ldots n\), and \(k=1,2,\ldots,R\).