## 1 Introduction  
The compressor is one of the most critical components in an aero engine, exhibiting viscous, compressible, and unsteady flow. In previous studies on transonic axial compressors [1], considerable efforts have been spent on improving the stage pressure ratio, efficiency, and stability margin. Historically, research about axial compressor design relied on empirical data correlations, and then the through-flow methods were proposed [2, 3]. Over the past two decades, computational fluid dynamics (CFD) has played a dominant role [3], while empirical information is still needed. Using CFD, researchers have efficiently analyzed the complex 3D compressor flow field [4]. A great deal of previous research has applied CFD tools with the Reynolds-averaged Navier-Stokes (RANS) equations [4, 5] because of the computational resource limitation. As a result of the computational capability improvement, several studies have emerged to evaluate the multistage and unsteady case [4]. A large and growing amount of the literature has investigated the optimization of compressors [6], including two main kinds of methods [7]: stochastic and gradient-based methods. The genetic algorithm (GA), a widely used stochastic method, has been successfully applied since the 2000s [8, 9]. Using the advanced NSGA-II algorithm [10], the pressure ratio, efficiency, and operating range of single-stage transonic axial compressorswere improved in several cases [11; 12]. [PERSON] et al. [13] compared particle swarm optimization (PSO) [14], GA, and a hybrid PSO-GA algorithm, then suggested that the hybrid PSO-GA algorithm performed best in a centrifugal compressor. In another aspect, several studies [15; 16] have used gradient-based algorithms in turbomachinery and focused on improving the efficiency and pressure ratio. Generally, better compressors can be found by optimization at certain constraints and objective settings; those studies outline a critical role for optimization in compressor design. One limitation of the above optimization methods is that their results cannot directly guide design; prior knowledge of the designers is needed, especially when the designs need to be modified frequently in a multidisciplinary or engineering context. Learning the design policy becomes possible with the development of reinforcement learning (RL) methods [17], which train the machine to learn the design like humans. In the past few years, RL methods have been used to successfully solve many complex design problems, including designing personalized therapies [18], designing proteins [19], and even finding matrix multiplication algorithms [20]. There are relatively few studies attempting to apply RL in aerodynamic design, focusing mainly on 2D airfoils for easier CFD analysis. Several deep RL algorithms [21] have been developed for a continuous state space, corresponding to the design space of compressors. Some authors have considered the usage of 2D external flow airfoils. A recent study by [PERSON] et al. [22] involved an application of proximal policy optimization (PPO) [23], which maximized the lift-to-drag ratio by exploring the design space. Similarly, [PERSON] [24] also used PPO to minimize the drag of a supercritical airfoil. In the turbomachinery field, internal flow airfoils have also been considered. [PERSON] [25] demonstrated that the total pressure loss could be reduced by modifying the 2D airfoil via a trained agent using a deep deterministic policy gradient (DDPG) [26]. These studies clearly indicated that RL methods could successfully learn the design policy of 2D airfoils and improve aerodynamic performance. However, no corresponding attempts for the 3D rotor case have been found to date. Different applications of machine learning methods exist in the literature regarding turbomachinery research. Much of the previous research on compressor optimization established the artificial neural network and its variations as surrogate models (such as [13; 16]). Another idea was proposed by [PERSON] et al.