# A Visual Analysis Method for Predicting Material Properties Based on Uncertainty
## 3 Performance-Prediction Model's Design
### Building a Performance Prediction Model
#### 3.2.1 Back-Propagation Neuron Network (BPNN) Model  
A BP neural network (BPNN) consists of three parts: the input layer, the hidden layer, and the output layer. The number of nodes in the hidden layer is the most critical factor affecting the capacity of the BPNN. In this study, the sigmoid function, the S-shaped transfer function, was used as the gradient function for the network. Additionally, we used the empirical formula Equation (1) to calculate this training. When the number of neurons in the hidden layer was three, the model's effect was relatively better, so the model with three layers of neurons in the hidden layer was used in this study, as shown in Figure 1.  
\[\text{T}(x)=\frac{1}{1+\exp(-x)} \tag{1}\]  
The choice of the number of nodes in the hidden layer is crucial for the construction of the model, which determines the performance of the model. We took a relatively small number of nodes in the hidden layer under the premise of satisfying the accuracy as much \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **Percentage of Data Volume** & **Influencing Factors** & **Predictive Performance** & **Amount of Data** & **Data Dimension** \\ \hline \hline \multirow{4}{*}{Data Set-1} & \multirow{4}{*}{50\%} & Response temperature & \multirow{4}{*}{Merny viscosity} & \multirow{4}{*}{4950} & \multirow{4}{*}{34} \\  & & PH regulator type and addition amount & & & \\  & & Stirring speed & & & \\  & & Monomer type & & & \\  & & Response temperature & & & \\ Data Set-2 & 100\% & PH regulator type and addition amount & & & \\  & & Stirring speed & & & \\  & & Monomer type & & & \\  & & Response temperature & & & \\ Data Set-3 & 50\% & & & & \\  & & Reaction pressure & & & \\  & & Reaction Kettle Types & & & \\  & & Response temperature & & & \\ Data Set-4 & 100\% & & Types of elicitor systems & & \\  & & Reaction pressure & & & \\  & & Reaction Kettle Types & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fluorinated-material dataset’s division. Figure 1: The input layer’s neurons are represented by light purple, blue represents the implicit-layer neurons, and green represents the output-layer neurons. as possible and used the empirical formula (Equation (2)) to show that the model's effect is comparatively better when the number of neurons in the hidden layer is three. This was verified in the later case study through the implementation of the experiment. \[M_{\text{hidden}}=\sqrt{M_{input}+M_{output}}+\gamma \tag{2}\]  
The number of neurons in the buried layer is denoted by \(M_{\text{hidden}}\); \(M_{input}\) and \(M_{output}\) represent the numbers of input and output neurons, respectively; \(\gamma\) represents a constant between 1 and 10. Therefore, In this research, a model (Figure 1) with three layers of neurons in the hidden layer was adopted. Calculated by the BP neural network algorithm, shown as a flowchart in Figure 1, the hidden layer's mth neuron receives the input layer's input, as shown in Equation (3), and the output layer's path neuron receives the input as shown in Equation (4). \[\beta_{n}=\sum_{n=1}^{k}\omega_{nm}z_{n} \tag{3}\]  
\[\partial_{p}=\sum_{m=1}^{n}\omega_{mp}z_{m} \tag{4}\]  
Orange indicates the error contained in the output layer, and the error is passed back to the implicit neuron through the inverse error function \(\text{E}(x)\), as shown in Equation (5) below. The weights and thresholds are adjusted according to the returned errors, and the cycle is continuously iterated so that the function avenue is minimized. \[\text{E}(x)=\frac{1}{2}{\sum_{i=1}^{p}{(\alpha_{i}-\beta_{i})^{2}}} \tag{5}\]  
\(\alpha_{i}\) is the desired output, and when we denote \(b-a\) as e, the inverse error function \(\text{E}(x)\) is expressed as shown in Equation (6). Then, we update the weight values as shown in Equations (7) and (8), where \(\eta\) is the learning rate.