## 1 Introduction  
Supercritical carbon dioxide (S-CO2) refers to the carbon dioxide fluid above the critical point (30.98 \({}^{\circ}\)C, 7.38 MPa) [1]. It has the advantages of being stable chemical properties, weak high-temperature corrosion, non-toxic, and non-combustible. At the same time, it has the characteristics of high specific heat capacity and density, high thermal conductivity, and low viscosity. S-CO2 is an ideal low-cost working medium [2; 3]. Therefore, a Brayton power generation system with S-CO2 has the advantages of high-efficiency, small-volume, and low-noise (mainly high-frequency noise). It has become one of the main research directions in the field of power generation technology (nuclear energy, solar energy, geothermal energy, waste heat, etc.) [4; 5; 6; 7]. Turbine is the \"heart\" of the whole power cycle. The power and efficiency of the system are directly affected by its performance. Therefore, the research on S-CO\({}_{2}\) turbine has become a hot spot. Sandia National Laboratory [8] developed a 100 kW centripetal turbine and a 50 kW centrifugal compressor, and conducted a large number of S-CO\({}_{2}\) closed cycle tests from 2007 to 2009. A labyrinth seal turbine wheel was developed by Korea Institute of Energy Research [9]. It was applied to a 10 kW S-CO\({}_{2}\) Brayton experimental loop. [PERSON] et al. [10] proposed a design method of S-CO\({}_{2}\) radial turbine. The one-dimensional (1-D) model and three-dimensional (3-D) numerical simulation methods were adopted to predict the off-design performance. [PERSON] et al. [11] completed the design of high-pressure and low-pressure axial-flow turbines applied to 5 MW S-CO\({}_{2}\) reheated Brayton cycle by using the self-designed program. The isentropic efficiencies of the turbines were 82.88% and 82.26%, respectively. A 10 MW S-CO\({}_{2}\) single stage centrifugal turbine was designed and numerically analyzed by [PERSON] et al. [12]. The total-static efficiency after blade shape optimization was 89.02%. At present, computational fluid dynamics (CFD) is still the main method of turbine aerodynamic design and analysis. However, a lot of iterative calculations are needed to solve the Navier-Stokes (NS) equation. This is time-consuming and expensive to calculate. It also delays the entire design and analysis cycle. Therefore, it is necessary to develop a more efficient and accurate method than CFD. With the development of computers, CFD method is widely used. Hence, a large number of CFD data are generated in the process of design and optimization. Therefore, the data-based proxy model becomes more and more practical and important. Previous studies have shown that when the machine learning algorithm is properly selected and fully utilized, surrogate models based on that can well predict the performance of components in power cycle. Based on Levenberg-Marquardt algorithm, [PERSON] et al. [13] proposed a back-propagation neural network to predict the off-design or overall dynamic performance of the gas turbine. [PERSON] and [PERSON] [14] developed a computational methodology based on artificial neural networks (ANNs). It could accurately predict the performance-curve and best-efficiency-point of turbo pump working in reverse mode. This proved that ANNs are a universal and effective evaluation tool. Based on neural network surrogate models, [PERSON] et al. [15] proposed an optimization model for main design parameters of the radial turbine. The designed neural networks had high accuracy and could accurately learn highly nonlinear physical model objects. [PERSON] et al. [16; 17; 18] developed the response surface methodology (RSM) for the optimization of a catalytic reforming micro-reactor and a thermosyphon heat pipe. The above examples have shown that machine learning can be used for component performance prediction, but such surrogate models belong to the black-box model, ignoring the physical relationship between parameters, and have little effect on grasping the operation rules of components and guiding component control.