## 2 Methods of BPNN-MDSG Modeling
The \"M\" and \"N\" on the summation symbol indicate that the target pipeline network has M inlets and N outlets. \(Q_{s}\) is the heat transfer between the pipeline and the external environment. After modular modeling of the parts, the parts are divided into pressure parts and flow parts. The two are used in turn, and their boundary conditions are mutually solving conditions, forming bidirectional parameter transfer. The MDSG system [13; 26] modeling method has been introduced in detail in previous studies, so it will not be repeated here. The BPNN-MDSG digital twin modeling method focuses on strengthening the connection between virtual space and real devices. By calibrating and optimizing the MDSG model of the compressor system through the historical data of the actual equipment, it can better complete the simulation and prediction of steady state, unsteady state, and the conditions without data (such as surge conditions). The BPNN-MDSG method aims to obtain the appropriate \(H\) and \(F\) for the compressor and \(C_{f}\) for the valve by training BP neural networks. BP neural network is a typical neural network method, which can theoretically realize the modeling of arbitrary nonlinear curves [27]. The three-layer forward network of the BP neural network basic structure is shown in Figure 1. It is composed of the input layer, hidden layer, and output layer. Where \(X=\{x_{1}\), \(x_{2}\), \(\ldots\), \(x_{i}\}\) is the amount of input to the network. \(Y=\{y_{1}\), \(y_{2}\), \(\ldots\), \(y_{i}\}\) is the hidden layer output of the neural network. \(Z=\{z_{1}\), \(z_{2}\), \(\ldots\), \(z_{i}\}\) is the output of the network. \(v_{ij}\) and \(w_{ij}\) are the weights corresponding to the neural connections between the input layer and the hidden layer, and the hidden layer and the output layer, respectively. \(\theta=\{\theta_{1}\), \(\theta_{2}\), \(\ldots\), \(\theta_{i}\}\) and \(\lambda=\{\lambda_{1}\), \(\lambda_{2}\), \(\ldots\), \(\lambda_{i}\}\) are the biases of the hidden layer and the output layer, respectively. In the forward transmission process of the signal, the output value of the previous layer of neurons is multiplied by the corresponding weight value of the neural connection and summed to obtain the input value of the hidden layer and the output layer. The neurons in the hidden layer and the output layer need to transform the input signal by an activation function. In this paper, the Sigmoid function was used as an activation function. The Sigmoid function is adopted as the activation function primarily due to its proven efficacy in modeling the graded, stage-sensitive characteristics of compressor performance curves, where system responses exhibit continuous transitions rather than binary thresholds. Its smooth nonlinearity enables precise approximation of complex thermodynamic relationships between input parameters and output targets. The Sigmoid function maps inputs to a continuous range of (0, 1), which is beneficial for modeling bounded relationships--such as the performance curves of compressors, where parameters often exhibit constrained, nonlinear behaviors within specific ranges. This aligns with the network's task of predicting parameters like \(H\) and \(F\) within narrow intervals [0.55, 0.555] and [28.5, 29.5], as Sigmoid helps constrain outputs to physically meaningful bounds. Take the signal transmission between the hidden layer and the input layer as an example. \[net_{\text{out}}=\sum_{j=1}^{j}y_{j}w_{jk} \tag{11}\]  
\[z_{k}=f(net_{\text{out}})=\frac{1}{1+e^{-(net_{\text{out}}+\lambda_{k})}} \tag{12}\]  
where \(net_{\text{out}}\) is the network input value of the \(k\)-th neuron in the output layer. In the process of error back propagation, the BP neural network calculates the error value between the output value and the target value, calculates the contribution of each neuron to the output error according to the chain rule, and transmits the error signal in  
Figure 1: Structure diagram of BP neural network. reverse.