## 5 Algorithm Design for Capacity Forecasting Driven by Twin Data
\(i_{t}\) can be expressed as shown in Equation (13):  
\[i_{t}=\sigma(W_{i}[h_{t-1},x_{t}]+b_{i}) \tag{13}\]  
Figure 3: LSTM neural network processing data process. where \(C_{t}^{\prime}\) can be expressed as shown in Equation (14):  
\[C_{t}^{\prime}=tanh\left(W_{c}\left[h_{(t-1)},x_{t}\right]+b_{c}\right) \tag{14}\]  
\(\Delta T_{t1}\), \(\Delta T_{t2}\), and \(\Delta T_{t3}\) are taken as the input signals and propagate forward along the network. Under the action of each neuron weight, \(w_{ij}\), and the activation function of \(g(e)\), the output value of the network coupled with three disturbance factors is obtained through the output layer. The input time series samples are considered as \(X= \begin{bmatrix}x_{11}&x_{12}&x_{13}\\ \cdots&\cdots&\cdots\\ x_{t1}&x_{t2}&x_{t3}\end{bmatrix} \) in the input layer, in which X is the three disturbance factors that affect the workshop production capacity. The output value is subtracted from the actual value to obtain network error b. This error is transmitted backward through the neural network, and the connection weights between neurons in each layer are corrected layer by layer. The hidden layers in the model are represented as shown in Equation (15):  
\[h_{tj}=\sigma(\sum_{i=1}^{n}x_{ti}u_{ji}+h_{t-1,j}w_{jj}) \tag{15}\]  
where \(x_{ti}\) is the value of the \(i\) node in the input layer at time \(t\), and \(u_{ji}\) indicates the weight coefficient matrix between the \(j\) node in the hidden layer and the \(i\) node in the input layer. In addition, \(h_{tj}\) represents the output value of the \(j\) node of the hidden layer at time \(t\), \(h_{t-1,j}\) defines the output value of the \(j\) node of the hidden layer at time \(t-1\), and \(w_{jj}\) symbolizes the weight coefficient matrix of the \(j\) node of the hidden layer during backpropagation. The number of hidden layer nodes in the model is set to m, and its output value is \(H= \begin{bmatrix}h_{11}&h_{12}&h_{1m}\\ \cdots&\cdots&\cdots\\ h_{t1}&x_{t2}&x_{tm}\end{bmatrix} \). The model output layer is shown in Equation (16):  
\[y_{t}=\sum_{j=1}^{m}\left(h_{tj}v_{kj}\right)+b \tag{16}\]  
where \(y_{t}\) is the output value of the output layer at time \(t\),\(V_{kj}\) indicates the weight coefficient matrix between the \(j\) node of the hidden layer and the \(k\) node of the output layer, and \(b\) denotes the bias term. The output layer of the model is shown in Equation (17):  
\[RMSE(y,y^{\prime})=\sqrt{\sum_{i=1}^{n}\frac{(y_{i}-y_{i}^{\prime})^{2}}{n}} \tag{17}\]  
where \(y\) represents the actual workshop production capacity, \(y^{\prime}\) indicates the predicted value output by the model, and \(n\) denotes the number of samples. It mainly needs to determine the five hyperparameters of the model, namely the time steps of the input layer, the dimension of the input layer, the number of hidden layers, the dimension of each hidden layer, and the dimension of the output variable. The LSTM neural network uses input information, \(x_{t}\), and \(h_{t-1}\) to form the final output objective function \(h_{t}\). By continuously iterating and updating the system, effective prediction results are formed. The iterative process of the LSTM neural network system is defined in Algorithm 1. ```
Input: Production capacity time series setting. Output: Time series value of the production forecast.