## 2 Structure of IT2 FNN  
Figure 1 depicts the structure of the IT2 FNN, which includes multiple inputs and a single output. The input of the network is denoted by X and output is denoted by Y. In traditional IT2 FS, the computational complexity in the defuzzification process is usually very high. Therefore, to increase the effectiveness and reduce complexity of the fuzzy system, this study used the type-reducer method, proposed by [PERSON] [12], in the defuzzification part. A functional link neural network (FLNN) with a  
nonlinear combined output was assigned a fuzzy rule in the consequent part [28]. Each fuzzy rule \(j\) in the IT2 FNN structure is expressed as follows:  
\[ \begin{array}{l}\text{Rule }j:\text{IF }x_{1}\text{ is }\widetilde{A}_{1j}\text{ and }x_{2}\text{ is } \widetilde{A}_{2j}\text{ }\ldots\text{ and }x_{i}\text{ is }\widetilde{A}_{ij}\text{, }\text{ and }x_{N}\text{ is }\widetilde{A}_{Nj}\\ \text{THEN }y_{j}\text{= }\sum_{k=1}^{M}\omega_{kj}\varphi_{k}\\ \text{= }\omega_{1j}\varphi_{1}+\omega_{2j}\varphi_{2}+\ldots+\omega_{Mj}\varphi_ {M}\end{array}  \tag{1}\]  
where \(i\) is the input number, \(x_{i}\) is the input variables, \(y_{j}\) is the local output variables, \(\widetilde{A}_{ij}\) is the type-2 fuzzy set, \(\omega_{kj}\) is the link weight of local output, \(\varphi_{k}\) is the basis trigonometric function of input variables, \(M\) is the basis function number, and rule \(j\) is the \(j\)th fuzzy rule. The proposed IT2 FNN has a five-layer NN. Here, \(u^{(l)}\) represents the \(l\)th layer output of a node, and each layer of a node operates as follows:  
**Layer 1**: This layer is the input layer, and each node is an input node. Only the input signal is passed to the next layer of the network. \[u_{i}^{(1)}=x_{i} \tag{2}\]
**Layer 2**: This layer is the membership function layer that performs the fuzzification operations. Each node is defined as a type-2 fuzzy set. Each membership function is a Gaussian membership function represented by an uncertain mean [\(m_{ij1}\), \(m_{ij2}\)] and a fixed standard deviation (STD) \(\sigma_{ij}\). The function is expressed as follows:  
\[u_{ij}^{(2)}=\exp\left(-\frac{\left[u_{i}^{(1)}-m_{ij}\right]^{2}}{\sigma_{ij} ^{2}}\right)\equiv N\left(m_{ij}\text{, }\sigma_{ij}\text{; }u_{i}^{(1)}\text{), }m_{ij}\in\left[m_{ij1}\text{, }m_{ij2}\right] \tag{3}\]  
where \(m_{ij}\) and \(\sigma_{ij}\) represent the \(i\)th input of the \(j\)-term mean and standard Gaussian membership function, respectively. The footprint of uncertainty of the Gaussian membership function represents a bounded interval of the upper membership function \(\overline{u}_{ij}^{(2)}\) and lower membership function \(\underline{u}_{ij}^{(2)}\). The output of each node in the second layer represents an interval \(\left[ \begin{array}{cc}\mathbf{u}_{\bar{\mathbf{j}}}^{(2)},&\mathbf{u}_{- \bar{\mathbf{j}}}^{(2)}\end{array} \right]\). The formula for calculating the membership degree is as follows:  
Figure 1: Structure of an interval type-2 fuzzy neural network (IT2 FNN). \[\overline{u}_{ij}^{(2)}\!\left(u_{i}^{(1)}\right)\!=\!\left\{ \begin{array}{ll}N \!\left(m_{ij1},\,\sigma_{ij};\,u_{i}^{(1)}\right)&if\,u_{i}^{(1)}<m_{ij1}\\ 1\,,&if\,m_{ij1}\leq u_{i}^{(1)}\leq m_{ij2}\\ N\!\left(m_{ij2},\,\sigma_{ij};\,u_{i}^{(1)}\right)&if\,u_{i}^{(1)}>m_{ij2} \end{array} \right. \tag{4}\]  
and  
\[\underline{u}_{ij}^{(2)}\!\left(u_{i}^{(1)}\right)\!=\!\left\{ \begin{array}{ ll}N\!\left(m_{ij2},\,\,\sigma_{ij};\,u_{i}^{(1)}\right),&if\,u_{i}^{(1)}\leq \frac{m_{ij1}+m_{ij2}}{2}\\ N\!\left(m_{ij1},\,\,\sigma_{ij};\,u_{i}^{(1)}\right),&if\,u_{i}^{(1)}>\frac{ m_{ij1}+m_{ij2}}{2}\end{array} \right.. \tag{5}\]  
**Layer 3**: This layer is also called the firing layer, and each node is a rule node; an algebraic product calculation is used to receive the firing strength of each rule node \(\overline{u}_{j}^{(3)}\) and \(\underline{u}_{j}^{(3)}\).