## 2 Methods of BPNN-MDSG Modeling  
The mathematical model of the compressor was derived from the laws of mass conservation, momentum conservation, and energy conservation. All the pressure and temperature in the text are absolute values. \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{1}\]  
\[\rho\frac{Du_{j}}{Dt}=-\frac{\partial p}{\partial x_{j}}+\mu\frac{\partial^{2 }u_{j}}{\partial x_{i}^{2}}+\rho f_{j} \tag{2}\]  
\[\frac{\partial(\rho h)}{\partial t}+\frac{\partial(\rho U_{x}h)}{\partial x} +\frac{\partial(\rho U_{y}h)}{\partial y}+\frac{\partial(\rho U_{z}h)}{ \partial z}=-pdivU+div(agradT)+\Phi+S_{h} \tag{3}\]  
where Equations (1) and (2) are tensor forms, and \(u_{i}\), \(x_{i}\), and \(f_{i}\) are the tensor forms of velocity, distance, and force, respectively. \(U_{x}\), \(U_{y}\), and \(U_{z}\) in Equation (3) are the velocities in the direction of \(x\), \(y\), and \(z\). The Greitzer model was proposed by [PERSON] in 1976 [20], which usually consists of the compressor, plenum, and throttle valve. In subsequent studies, [PERSON] and [PERSON] [21,22] described the dynamic characteristics and surge characteristics of compressors based on these three parts. The MDSG model and compressor system dynamic modeling methods were proposed in our previous research, which can not only set the characteristic parameters of target components specifically, improving the simulation accuracy, but also conform to the actual structure of the complex system. MDSG model divides the component modules into two categories. The input of one type of module is pressure, the output is mass flow rate, and the other is vice versa. The two modules appear alternately to realize the system simulation. The mathematical equations for each component are given in turn below. Based on the Greitzer model [20] and MDSG model, the dimensionless performance map of the compressor and isentropic thermodynamics model is developed. The optimized equations are as follows [13; 26]. \[\frac{dp_{\rm c}}{dt}=\frac{{a_{\rm 01}}^{2}}{V_{\rm c}}(m_{\rm in}-m_{\rm out}) \tag{4}\]  
\[\frac{dm_{\rm c}}{dt}=\frac{A_{\rm c}}{L_{\rm c}}(\psi(\frac{N}{\sqrt{T_{\rm in }}},\frac{m_{\rm c}\sqrt{T_{\rm in}}}{p_{\rm in}})p_{\rm in}-p_{\rm c})-\frac{ \lambda R_{g}}{D_{h}A_{\rm c}}\frac{m_{\rm out}^{2}T_{\rm out}}{p_{\rm c}+p_{ \rm out}}-\frac{R_{g}}{A_{\rm c}L_{\rm c}}(m_{\rm out}^{2}\frac{T_{\rm out}} {p_{\rm out}}-m_{\rm in}^{2}\frac{T_{\rm in}}{p_{\rm in}}) \tag{5}\]  
where \(m_{\rm c}\) is the mass flow rate of compressor, \(a_{\rm 01}\) is the voice speed, and \(\Psi\) is the dimensionless performance function of the compressor. \(\lambda\) is the flow coefficient; its definition is as shown in Equation (6). \(D_{h}\) is the equivalent hydraulic diameter of the compressor impeller passage. \(A_{\rm c}\) and \(L_{\rm c}\) are the cross-sectional areas and effective length of the equivalent compressor duct, respectively. \(V_{\rm c}\) is the volume of the internal flow passage of a single compressor. \(p_{\rm c}\) is the lumped pressure of the internal flow passage volume \(V_{\rm c}\). The subscripts \"out\" and \"in\" define the outlet and inlet of the compressor, respectively. Equation (4) calculates the pressure change trend of the compressor's volume based on the changes in mass flow rates entering and leaving the compressor. The first term on the right side of Equation (5) is mainly used to calculate the trend of the change in compressor flow. Firstly, the target pressure was calculated by using the compressor inlet pressure and the performance curve. Then, the difference between the target pressure and the current compressor volume pressure \(p_{\rm c}\) is obtained, which represents the trend of the compressor pressure approaching the performance curve. The second and third terms on the right side are for correction calculations to improve accuracy. The second term is used to eliminate the loss caused by the matching degree between the impeller speed and the outlet flow.