## 3 Overall Architecture of the Proposed Method
Second, the original input values of \(T(k)\) are increased to \(T(k-i)\) by 10% and decreased by 10% one by one, and the original values of the other input items are kept unchanged at the same time. This results in new \(i\) datasets. The new dataset is substituted into the trained network to obtain network outputs \(D_{i+}\) and \(D_{i-}\). Finally, we use \(D_{i+}\) and \(D_{i-}\) to make a difference, and the result is the impact value (IV). The MIV of this input is obtained by calculating the average of IV. The \(|MIV|\) corresponding to the \(i\)-th input item is as follows:  
\[MIV=\frac{1}{M}\sum_{j=1}^{M}(D_{i+}-D_{i-}) \tag{3}\]  
All \(|MIV|\) from \(T(k)\) to \(T(k-i)\) are calculated. The larger \(|MIV|\) is, the greater the impact of this input on the output \(D(k)\) of the network. \(\delta_{t}\) is the temperature threshold. Finally, we select all inputs \(T(k-n)\) that satisfy \(|MIV|>\delta_{t}\), where \(n\) is the delaying order. Similarly, the delaying order \(m\) of the pulling speed is also obtained by this method. After determining the delaying order of the influence of the temperature and pulling speed on the crystal tail diameter, a neural network model is established to predict the diameter. The model also needs to introduce the crystal length and the crystal mass as input variables. The tail length is introduced considering that the thermal radiation will change as the crystal length increases [3], and the introduction of a mass increment indirectly reflects the change in diameter. The model can be expressed using Formula (4):  
\[D(k)=F(T(k-l),\cdots,T(k-n),V(k-p),\cdots,V(k-m),L(k),M(k)) \tag{4}\]  
where \(T(k-l),\cdots,T(k-n)\) are satisfied temperature inputs that meet \(|MIV|>\delta_{t}\), \(V(k-p),\cdots V(k-m)\) are satisfied pulling speed inputs that meet \(|MIV|>\delta_{v}\), \(\delta_{v}\) is the pulling speed threshold, \(L(k)\) is the crystal tail length at the time of \(k\), \(M(k)\) is the mass increment from the time \(k-1\) to \(k\), and \(F(\cdot)\) is a nonlinear function. To improve the accuracy of the tail diameter prediction, it is necessary to fully extract the characteristic information in the data. Although several soft measurement models based on deep learning can achieve this goal, those models still have limitations. For  
Figure 5: Three-layer multi-input and single-output temperature-diameter dynamic neural network model. example, in the case of a large amount of data, complex physical structures, and increased dimensions, the SVR modeling method is generally inefficient [15; 16]. Also, this method only uses a shallow network for training rather than repeatedly training through a multi-layer network, so it cannot extract the characteristic information in the data, which will affect the model's prediction accuracy. In addition, the BP neural network has a higher efficiency in a single hidden layer. When there are multiple hidden layers, the efficiency of back propagation will be greatly reduced. Therefore, this paper uses a DBN with multiple hidden layers for soft measurement modeling [17; 18]. A DBN can learn complex data features by combining multi-layer networks to realize multi-level representation learning and represent the implicit information of data, thus making the prediction more accurate. The DBN is composed of multiple restricted Boltzmann machines (RBM) [19]. The structure of the DBN model is shown in Figure 6. An RBM visible layer is used to describe the features of the input data, and the hidden layer can be regarded as a feature-extraction layer. The visible layer unit is \(v=\{v_{1},v_{2},\cdots,v_{i}\}\), the hidden layer unit is \(h=\{h_{1},h_{2},\cdots,h_{j}\}\), and the internal parameter set is \(\theta=\{w,a,b\}\), and then the probability distribution based on the energy probability model is as follows:  
\[P(v,h;\theta)=\frac{1}{Z(\theta)}e^{-E(v,h;\theta)} \tag{5}\]  
where \(Z\) is the regularization factor and \(Z(\theta)=\sum_{o}\sum_{h}e^{-E(v,h;\theta)}\).