# Optimizing Generative Adversarial Network (GAN) Models for Non-Pneumatic Tire Design
## 2 Test Methods
### Projected GAN  
Projected GAN adds a feature projector, \(\mathrm{P}_{\mathrm{I}}\), to the original GAN, as shown in Equation (8). \(\mathrm{P}_{\mathrm{I}}\) maps the real image and the generated image into the input space of the discriminator. \[\min_{\mathrm{G}}\max_{\{\mathrm{D}_{\mathrm{I}}\}}\sum_{\mathrm{I }\in\mathrm{L}}(\mathbb{E}_{\mathrm{x}}[\mathrm{logD}_{\mathrm{I}}(\mathrm{P }_{\mathrm{I}}(\mathrm{x}))]+\mathbb{E}_{\mathrm{z}}[\mathrm{log}(1-\mathrm{D }_{\mathrm{I}}(\mathrm{P}_{\mathrm{I}}(\mathrm{G}(\mathrm{z}))))]) \tag{8}\]  
\(\mathrm{D}_{\mathrm{I}}\) works on any independent feature projection. \(\mathrm{P}_{\mathrm{I}}\) is fixed, and only the parameters of \(\mathrm{G}\) and \(\mathrm{D}_{\mathrm{I}}\) are optimized. By fixing \(\mathrm{P}_{\mathrm{I}}\), no competition among the producers occurs. Thus, the variation in the producers is small, resulting in stable learning. The projected GAN proposed three main methods to speed up the convergence of learning to make it comparable to that of existing GANs and solve the problem of the discriminator not fully utilizing the pre-trained features of deeper layers. Multi-scale discriminators: Multi-scale discriminators extract features from four layers, \(\mathrm{L}_{\mathrm{I}}\), of a pre-trained feature network F. Each layer, \(\mathrm{L}_{\mathrm{t}}\), is applied to a separate discriminator, \(\mathrm{D}_{\mathrm{I}}\), using a simple convolutional architecture that uses spectral normalization [48]. We used a simple convolutional architecture, where each discriminator \(\mathrm{D}_{\mathrm{I}}\) utilizes spectral normalization [48]. We found that the best performance was achieved by applying convolution until each discriminant had a scale of 4, 4 and then selecting logit. Finally, the logit values generated by each of the four discriminators were summed and used as the loss. Random projections: We identified two strategies to make information about all features available to the discriminator. The first is cross-channel mixing (CCM) and the second is cross-scale mixing (CSM). CCM is a cross-channel mixing method that applies a \(1\times 1\) convolution with the same number of inputs, outputs, and channels so that the input is more informative. Increasing the number of output channels improves performance because the image information can be better preserved. CSM uses \(3\times 3\) convolution and  
Figure 8: Comparison between real images and images generated by StyleGAN v2-ADA. bilinear interpolation to mix the CCM output with the CCM output. These two methods allow us to preserve large amounts of information with fewer layers and the performance is better than that of traditional GANs. Pre-trained feature network: In this scheme, we experimented with different feature networks. The first, EfficientNets [49], allows for direct control of model size and performance; that is, by changing or removing the structure of the model, the size of the model can be reduced or increased, and the performance of the model changes accordingly. EfficientNets is an image classification model trained with ImageNet [50] that models the tradeoff between accuracy and computation. Second, we used ResNets [51] of different sizes. We also used ResNets trained on the R50-CLIP dataset, which is a dataset of image-text pairs, instead of ImageNet, to analyze how models pre-trained with features from ImageNet affect GAN. Finally, we used the vision transformer architecture (ViTBase) [52] and its successor (DeiT small distilled) [53]. We excluded the inception network [54] because it is used to extract feature maps from FID metrics, and the generated images can be correlated with FIDs. Unlike traditional GAN models, Projected GAN is a method that trains a GAN model using a pre-trained situation. This increases data efficiency and removes unnecessary constraints to improve image quality, training speed, and sample utility.