## 5 Algorithm Design for Capacity Forecasting Driven by Twin Data  
As can be seen in the previous section, a digital twin system is established before constructing a new factory and processing new products based on the construction of high-fidelity simulation models and the integration of constraints such as production factors, production plans, and production process control. Through the digital twin system, various data can be collected to evaluate the production capacity of new factories and new product processing. This prediction method can effectively address problems such as inaccurate predictions and the inability to meet delivery dates due to insufficient historical data and personnel experience. Production capacity analysis based on discrete manufacturing workshops involves the prediction of future production capacity based on twin data operations. Since the digital twin data belong to time series data, after analysis, this paper uses the short-term memory network algorithm of recurrent neural networks (RNNs) to predict production capacity. LSTM is a temporal recurrent neural network designed to solve the long-term dependency problem of general RNNs, which can address the problems of gradient vanishing and gradient explosion in RNNs. The unique design structure of LSTM makes it suitable for processing and predicting important events with very long intervals and delays in the time series. Recurrent neural networks have the form of repeated module chains of neural networks. In the LSTM chain structure, each module contains four network layers, as shown in Figure 3. Additionally, LSTM has three gates (information-forgetting gate, input gate, and output gate) to optimize neuronal information. The output of the result can be expressed as Equation (7):  
\[h_{t}=o_{t}\times tanh(C_{t}) \tag{7}\]  
where \(o_{t}\) is used to determine the output gate information transmitted by the neurons. From the input \(h_{t-1}\) of the previous neuron and the raw input, \(x_{t}\) is calculated by the activation function and weight parameters. It can be expressed as shown in Equation (8):  
\[o_{t}=\sigma(W_{o}[h_{t-1},x_{t}]+b_{o}) \tag{8}\]  
where \(W_{o}\) and \(b_{o}\) are weight parameters and paranoia parameters, respectively. In Equation (7), \(\tan h\) is the activation function with an output range of \([-1,1]\). Moreover, the value range of the derivative function is 0-1, which can effectively solve the problem of gradient disappearance in the cyclic network. The formula can be expressed as shown in Equation (9):  
\[\tan h(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \tag{9}\]  
In Equation (8), \(\sigma\) represents the activation function, which is a smooth step function and differentiable. It can convert any value between 0 and 1 for binary classification using Equation (10). \[\sigma=\frac{1}{1+e^{-x}} \tag{10}\]  
\(C_{t}\) in Equation (7) represents the updated information of neurons, which effectively eliminates the long-term dependency problem in neural networks and establishes a more accurate connection between input and output information. The parameter \(C_{t}\) can be expressed as shown in Equation (11):  
\[C_{t}=f_{t}\times C_{t-1}+i_{t}\times C_{t}^{\prime} \tag{11}\]  
where \(f_{t}\) indicates the forgetting gate information, which determines the information that must be discarded in the network state. In a sequence of data, different data may change in focus over time, forgetting unnecessary and unimportant information, which is also the reason for the emergence of forgetting gates. \(f_{t}\) can be expressed as shown in Equation (12):  
\[f_{t}=\sigma\Big{(}W_{f}[h_{t-1},x_{t}]+b_{f}\Big{)} \tag{12}\]  
where \(i_{t}\) and \(C_{t}^{\prime}\) are input gate information. In the process of neuron update, the first step is to delete information, followed by the update of information. The network update information is obtained by transforming the previous neuron input \(h_{t-1}\) and system input \(x_{t}\) through the activation function of \(\sigma\) and \(\tan h\).