## 1 Introduction
### Related Works  
#### 1.1.1 Generative Adversarial Networks  
Deep learning has been widely used in fields such as classification, facial recognition, and medical imaging. With the development of deep learning, many scholars have proposed various algorithms based on CNN. Recently, [PERSON] et al. (2024) [20] proposed a framework based on a generative adversarial network (RIW-GAN) for robust image watermarking. The encoder network comprises convolutional layers, and a residual block generates an encoded image that closely resembles the original with minimal distortion. A simulated noise layer, used as a differentiable network layer, enables end-to-end training before decoding, enhancing resilience against attacks. Experimental results demonstrate that RIW-GAN surpasses state-of-the-art models in terms of invisibility and robustness, effectively withstanding standard assaults like JPEG compression and geometric attacks such as resizing and cropping, thereby establishing its superiority in robust image watermarking. In 2020, [PERSON] et al. [21] proposed a generative adversarial network-based robust image watermark algorithm. The model comprises generator and adversary modules. The generator creates watermarked images and decodes noise-damaged images to retrieve the watermark, while the adversary identifies the presence of a watermark and introduces noise. A high-pass filter is applied before the discriminator, embedding the watermark in the mid-frequency image region based on the Hidden model. To calculate loss, a higher weight is assigned to the image center and a lower weight to the edges, aligning with the human visual system. Experimental results show superior visual performance and robustness against noise interference. [PERSON] et al. (2014) [PERSON] et al. (2014) proposed the architecture of generative adversarial networks (GAN). A GAN is an unsupervised learning method whose architecture consists of a generator and a discriminator. The generator network takes random noise as input, and its output generated pictures should be as similar as possible to the real samples provided by the training set. The input of the discriminator network is the real sample and the output of the generating network, and its purpose is to distinguish the output of the generator network from the real sample as much as possible. The two networks are trained against each other, and the ultimate goal is to prevent the discriminator network from correctly judging whether the generated image is true as much as possible. In recent years, many GAN-based architectures have been proposed and applied in many fields. The original GAN only needs a random noise as input to start training. In other words, the appearance of the generated image is completely determined by the random noise. The entire training process is too free and uncontrollable. To solve this problem, [PERSON] et al. (2014) [PERSON] et al. (2014) proposed the conditional GAN, which uses random noise and a piece of additional information as input at the same time. The additional information can be categories, images, text, etc. This change makes the training process controllable. In other words, a conditional GAN transforms GAN from unsupervised learning to supervised learning, allowing researchers to better control the training process. In 2015, [PERSON] et al. (2015) proposed deep convolutional generative adversarial networks (DCGANs) [PERSON] et al. (2015), which introduced convolutional neural networks (CNN) into the GAN architecture and improved the stability of training and the quality of the generated images. In the DCGAN architecture, the generator uses transposed convolution to replace the fully connected layer, and the discriminator uses the convolution layer to improve the image feature extraction. The discriminator uses Leaky ReLU instead of ReLU to avoid the gradient being too sparse (Dead ReLU Problem). Although DCGANs improve training stability and image quality, they still do not solve the two problems of GAN: the vanishing gradient problem and model collapse.