## 2 The Proposed Model and Strategies
### The Proposed Fractional Discrete Calculus Neural Network Modeling Approach  
Considering the important role of ANN models in the product form design process, the development of more flexible models is attracting increasing attention. The goal of this paper is to propose a discrete fractional-order modeling approach in order to generalize the existing integer-order models. To this end, the necessary background from discrete fractional calculus [62; 63] will be presented here. Figure 3: A visualized flowchart of the neural network model (1). **Definition 1** ([29; 62]). : _For \(\alpha\in\mathbb{R}\) and \(\eta\in\mathbb{R}-\{\ldots,-2,-1,0\}\), the rising factorial of \(\eta\) is given by_  
\[\eta^{\alpha}=\frac{\Gamma(\eta+\alpha)}{\Gamma(\eta)}\]  
_with \(0^{\alpha}=0\), where \(\Gamma\) is the Gamma function._  
**Definition 2** ([28; 29]). : _For \(0<\alpha<1\), the Caputo nabla fractional difference for the function \(\varphi:\mathbb{N}_{1}\to\mathbb{R}\) of order \(\alpha\) is given as_  
\[{}^{C}\
abla^{\alpha}_{0}\varphi(\eta)=\frac{1}{\Gamma(1-\alpha)}\sum_{s=1}^{ \eta}(\eta-s+1)^{-\alpha}\
abla\varphi(s),\ \eta\in\mathbb{N}_{1}.\]  
Using the above fractional discrete operator, model (1) can be extended to the following discrete fractional-order neural network model  
\[{}^{C}\
abla^{\alpha}_{0}\mathcal{Z}_{h}(\eta)=-a_{h}\mathcal{Z}_{h}(\eta)+ \sum_{j=1}^{n}w_{hj}f_{j}(\mathcal{Z}_{j}(\eta))+\Theta_{h},\ \eta\in\mathbb{N}_{0}, \tag{3}\]  
where the fractional order \(0<\alpha<1\), \(\mathcal{Z}_{h}(\eta)\), represents the state at \(\eta\); the parameters \(a_{h}\), \(w_{hj}\), \(f_{j}\), \(\Theta_{h}\) are as in (1); and we consider the initial conditions as \(\mathcal{Z}_{h}(0)=\mathcal{Z}_{h}^{0}\in\mathbb{R}\), \(h=1,2,\ldots,n\). **Remark 1**. : _In this paper, the fractional calculus approach is used to generalize some ANN models proposed in industrial product form design. The proposed discrete fractional-order model (3) extends the existing integer-order models [5; 6; 7; 8; 9; 10; 11] to the fractional-order case. Compared to other existing works, the use of the fractional-order operator in (3) overcomes the limitations of using integer-order differences. Additionally, the proposed fractional-order model (3) can best describe fractal and more undetermined forms [14; 31], rather than integer-order ANN models. As a special case, if \(\alpha=1\) holds, then model (3) is reduced to an integer-order model of type (1)._  
**Remark 2**. : _Different from the fractional-order model applied in [31], the nabla fractional operator is used in (3). Hence, the proposed model and established results contribute to the development of the field and provide more opportunities for applied researchers to adopt the fractional-order modeling approach that best suits the respective applied problem._  
In the practical stability analysis of the introduced model, the following properties of the fractional Caputo nabla difference will be used [25; 26; 28; 29; 62; 63]. **Lemma 1** ([25; 28]). : _If \(\varphi:\mathbb{N}_{0}\to\mathbb{R}\) and \(0<\alpha<1\), then_  
\[{}^{C}\
abla^{\alpha}_{0}\varphi^{2}(\eta)\leq 2\varphi(\eta)^{C}\
abla^{ \alpha}_{0}\varphi(\eta),\ \eta\in\mathbb{N}_{0}.\]  
**Lemma 2** ([25; 29]). : _If \(\varphi:\mathbb{N}_{0}\to\mathbb{R}\) and \(0<\alpha<1\), then_  
\[{}^{C}\
abla^{\alpha}_{0}|\varphi(\eta)|\leq sign(\varphi(\eta))^{C}\
abla^{ \alpha}_{0}\varphi(\eta),\ \eta\in\mathbb{N}_{0}.\]  
Moreover, the nabla discrete Mittag-Leffler function, defined by [28]  
\[\mathbb{E}_{\gamma_{1},\gamma_{2}}(\lambda,\zeta)=\sum_{l=0}^{\infty}\lambda^ {l}\frac{\zeta^{l\gamma_{1}+\gamma_{2}-1}}{\Gamma(\gamma_{1}l+\gamma_{2})}\]  
for \(\lambda\in\mathbb{R}\), \(|\lambda|<1\), \(\gamma_{1},\gamma_{2},\zeta\in\mathbb{C}\) with \(Re(\gamma_{1})>0\), will be used in the subsequent analysis.