## 1 Introduction  
With the rapid advance in deep learning techniques and access to vast amounts of data, generative models have come a long way in recent years. Generative Adversarial Networks (GANs) [1, 2, 3] belong to a powerful subclass of generative models. They work by pitting two networks against each other in a game-like scenario. The generator network creates synthetic data from a noise source, while the discriminator network distinguishes between the generator's output and real data. Notably, these models can produce visually stunning outputs without explicitly computing the probability densities of the underlying distribution. Due to their ability to learn representations, GANs have been utilized in various fields, including data synthesis [4, 5], semantic image editing [6], style transfer [7], image super-resolution [8] and classification [9]. However, achieving consistent and stable GAN training remains an ongoing challenge. Despite this, GANs often encounter another issue of mode collapse. This means that they tend to generate samples with limited diversity, even if they are trained on a dataset that is quite varied. In this paper, we demonstrate specific strategies to stabilize GAN training, which lead to higher-quality image generation and potentially mitigate the mode collapse issue. In Section 4, we introduce the SoftGAN, a novel approach inspired by fuzzy set theory [10] and fuzzy concept modeling [11]. Our goal for the discriminator is to establish fuzzy concepts of real data, where the boundary between real and generated data is as soft as possible. To achieve this, we balance the principles of maximum concept coverage and the principle of maximum expected entropy of fuzzy concepts, as proposed in [12]. During the early stage of SoftGAN training, the principle of maximum expected entropy dominates the learning process due to the considerable divergence between the generated and real distributions. As the training proceeds, the principle of maximum concept coverage takes over, as the divergence between the two distributions diminishes. By constantly adjusting the discriminator's tolerance based on the generator's ability, we implicitly drive the generated distribution closer to that of real data while avoiding the issue of gradient vanish. In total, our contributions to the GAN-based generative models field are three-fold:  
* Mitigating mode collapse and enhancing training stability through the dynamic borderline softening mechanism. * Proposing a novel perspective of GAN training through the learning of fuzzy concepts. The discriminator aims to learn a fuzzy concept of real data with a soft borderline between real and generated data, achieving a balance between maximum concept coverage and the maximum expected entropy of fuzzy concepts. * Offering theoretical and practical advancements in GAN training. Empirical evidence showcases the superiority of the SoftGAN in addressing mode collapse and generating higher-quality outputs compared to existing approaches. The presented approach possesses numerous advantages when compared to state-of-the-art GAN-based generative models, as outlined below:  
* The incorporation of the dynamic boundary softening mechanism enhances training stability and directs the generator to navigate through the entire training process without becoming ensnared in partial modes. The effectiveness of the SoftGAN in addressing mode collapse issues becomes evident through the examination of the Geometry Score indicator [13], as discussed in Section 6.3. * Unlike WGAN-based methods [3; 14; 15; 16] that focus on finding a discriminator that satisfies Lipschitz constraints to mitigate gradient vanishing, the SoftGAN adjusts the discriminator smoothness using the dynamic boundary softening mechanism. This prevents gradient vanishing while maintaining the convergence speed. Simultaneously averting gradient vanishing and sustaining the convergence speed, this mechanism ensures a balanced optimization process.