## 1 Introduction  
Adaptive optics systems use control algorithms to correct for aberrations and measure phases in real time with a wavefront sensor to ensure the optical system is immune to interference and adaptable to the environment; therefore, they have been widely used in laser communications, femtosecond lasers, and astronomical observations [1, 2, 3, 4, 5, 6, 7]. Currently, Shack-Hartmann wavefront sensors are the most effective and popular wavefront sensors [8]. However, unfavorable conditions, such as a weak target light source, an uneven light intensity distribution, and a high large light amplitude, result in a threshold value that is too low for the wavefront to interpret an accurate, leading below which the wavefront sensor extracts the effective signal leads to a significant reduction in the measurement accuracy that affects the dynamic correction performance of adaptive optics [9, 10]. In contrast, a wavefront sensorless system has a higher accuracy than systems with wavefront sensors when the target light energy is weak, and their simple construction makes adaptive optics more affordable than other solutions under certain circumstances [11]. In general, wavefront sensorless systems are divided into two main categories: model-based and model-free correction algorithms. Model-based optimal control algorithms include the mode method and the optical principle method [12, 13, 14]. Theoretically, model-based algorithms need to calculate and measure the relevant parameters of a complex system model, which is considerably complex. The two most common types of model-freecorrection algorithms are optimization algorithms and neural networks. The optimization algorithms predominantly include the stochastic parallel gradient descent algorithm (SPGD), the particle swarm optimization algorithm (PSO), and the simulated annealing algorithm. Such algorithms calculate the system evaluation function using the far-field light intensity data and iterate continuously until the evaluation function reaches the optimal point of the system, thereby achieving the wavefront correction [15; 16]. However, the optimization algorithm requires multiple iterations. Furthermore, as the system dimension increases, the convergence speed slows, making it difficult to meet the high demands of real-time system performance at a high resolution system requirements. Nevertheless, the neural network class only needs to extract the image data once to complete the wavefront correction, which improves its real-time performance of the wavefront correction. [PERSON] [17] proposed the adoption of the Inception v3 network to predict the phase information. The input of a CNN was the far-field image, and the output was the Zernike coefficient. This coefficient was employed to generate the phase information without iterations. [PERSON] et al. [18] changed the idea of the output Zernike coefficient and employed the U-Net network to directly predict the phase map of the wavefront. [PERSON] et al. [19] ameliorated an SH-Net on the basis of ResNet and U-Net results. The inputs of the network were the data collected by the Hartmann sensor, and the output was the wavefront phase image. [PERSON] et al. adopted EfficientNet-B0 [20] to predict sixty-six Zernike coefficients [21]. Nevertheless, these algorithms still needed to calculate the correction voltage using a wavefront correction algorithm with the phase information. These networks, such as U-Net and Resnet, are excellent in the field of image classification. [PERSON] [22] proposed a method using reinforcement learning, which uses a model-free reinforcement learning algorithm to optimize the recurrent neural network controller, which could effectively learn the random disturbances caused by the telescope's own vibration, hysteresis, and atmospheric disturbances, and it continuously adapted to the external environment.