## 2 Methods of BPNN-MDSG Modeling
The weights and biases between corresponding neurons were updated by the gradient descent method. Iterate until the error meets the output requirements. \[E=\frac{1}{2n}\sum_{l=1}^{N}\sum_{k=1}^{M}\left(O_{k}-y_{k}\right)^{2} \tag{13}\]  
where \(O_{k}\) is the actual value of the \(k\)-th neuron in the output layer. \(y_{k}\) is the predicted value of the \(k\)-th neuron in the output layer. Figure 2 shows the training of compressor module using BP neural network in BPNN-MDSG model. The input parameters are compressor flow rate, inlet pressure, outlet pressure, inlet temperature, outlet temperature, and compressor speed. The key characteristic parameters \(H\) and \(F\) in the dimensionless performance curve of the compressor are taken as outputs. This study configures 10 hidden neurons based on rigorous empirical validation. The compressor system exhibits strong nonlinear dynamics. A \"6-10-2\" architecture of BPNN optimally balances model capacity and computational efficiency. Table 1 presents the specific training and performance parameters of the neural network. Figure 3a,b, respectively, show the average variance loss and the distribution of prediction errors of the neural network. The \"6-10-2\" architecture with Sigmoid hidden-layer activation achieved a final MSE of 7.53 \(\times\) 10\({}^{-5}\) after 711 training epochs. This performance slightly exceeds the target MSE 1 \(\times\) 10\({}^{-5}\) while demonstrating computational efficiency with a 35 s training time. A total of 58,079 sets of data were collected from actual ATF compressor systems, and the data partitioning followed rigorous standards: 75% training (43,559 samples), 15% testing (8712 samples), and 10% validation (5808 samples). The model effectively balanced expressiveness against computational constraints during optimization. In Figure 3, the trends of the MSE (Mean Squared Error) losses for training, validation, and testing were the same, and the target accuracy was achieved after 711 rounds. The network's shallow architecture does not suffer significantly from the Sigmoid function's known limitation of vanishing gradients. In shallow networks, Sigmoid function gradient retains sufficient magnitude to propagate errors effectively, as evidenced by the network's ability to learn from 43,559 training samples and generalize to 8712 testing samples with low loss. Furthermore, the distribution of prediction errors meets the requirements of prediction accuracy. The BPNN did not exhibit overfitting or underfitting even at a learning rate of 10%. Figure 2: The structure diagram of BPNN in MDSG modeling. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline Network Architecture & 6-10-2 & Input-hidden-output layers \\ Activation Function & Sigmoid (Hidden) & Nonlinear transformation \\ Training Epochs & 1000 (max)/711 & Actual trained epochs \\ Target MSE Loss & \(1\times 10^{-5}\) & Mean squared error \\ Final MSE Loss & 7.5288 \(\times\)\(10^{-5}\) & Mean squared error \\ Learning Rate & 0.10 & Gradient update step size \\ Training Set Size & 43,559 (75\%) & Samples for model training \\ Test Set Size & 8712 (15\%) & Samples for performance evaluation \\ Validation Set Size & 5808 (10\%) & Samples for performance validation \\ Training Time & 35 & Seconds \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The training and performance parameters of the BPNN. Figure 3: The structure diagram of BPNN in MDSG modeling: **(a)** The MSE loss of BPNN. **(b)** The prediction error distribution. In the iteration of the neural network, the dimensionless performance curve was obtained from the \(H\) and \(F\) of the iteration. Secondly, the flow rate, temperature, and pressure of the compressor were obtained through the performance curve. Finally, an iteration was completed by calculating the error transfer of flow, temperature, and pressure and checking the neural network. Then the BP neural network was constructed, and the final parameters \(H\) and \(F\) were obtained through training.