## 2 Materials and Methods
### Models and Methods
#### 2.3.2 cGAN Architecture  
The Generative Adversarial Network (GAN) is a deep neural network type that generates new data based on the data statistics of the training set [31]. GANs have two essential components, the generator and the discriminator, which are trained against each other using game theory. The generator produces candidates that the discriminator evaluates. Though GANs were originally developed for unsupervised learning, incorporating labels as constraints can result in a subcategory known as conditional GANs (cGANs), as depicted in Figure 4. Our research focuses on developing a deep learning model that utilizes a conditional generative adversarial network (cGAN) and paired image data [32; 33]. The cGAN model comprises two crucial components: the generator, known as U-Net, and the discriminator, referred to as PatchGAN [32]. The role of the generator is to take geometric images (labels or constraints) as input and produce field images of interest by incorporating random noise. Subsequently, the discriminator compares these generated field images to authentic images obtained from finite element modeling (FEM). The generator's objective is to increase the error rate of the discriminator. In contrast, the discriminator aims to optimize its ability to distinguish between fake images generated by the generator and real ones. To accomplish this, we utilize TensorFlow [30], a versatile machine learning framework, to perform calculations and implement the generative adversarial network (GAN) architecture for translating arterial wall geometries into strain fields [32]. Our specific model employs U-Net as the generator and PatchGAN as the discriminator. U-Net generates synthetic strain field images based  
Figure 4: Workflow for the proposed method begins with a random generator using Python scripting to produce 2D arterial wall geometry images (256 \(\times\) 256). Next, a FEM analysis is conducted to obtain arterial walls’ true strain field information under blood pressure. ML model called Conditional Generative Adversarial Network (cGAN), which includes a generator U-Net and a discriminator PatchGAN, is trained to predict strain fields from geometry images. The generator generates strain field maps using the geometry images as input. The discriminator then compares these generated images with real FEM-derived images. A well-trained model can accurately predict strain field maps, validated against high-fidelity FEM models and arterial walls with new geometries. on arterial wall geometries, and PatchGAN evaluates the authenticity of these generated field images by comparing them to real strain field images. Our generator U-Net shares similarities with the CNN models discussed earlier, and the generator loss function is defined as:  
\[Gen_{loss}=\lambda\times gan_{loss}+L_{1}\_loss \tag{2}\]  
The variable _gan_loss_ represents the sigmoid cross-entropy loss between the generated images and an array of ones. At the same time, \(L_{1}\_loss\) calculates the mean absolute error between the generated and target images. It should be noted that (2) is equivalent to (1) when \(\lambda\) is set to 0 and \(L_{1}\_loss\) is replaced by \(L_{2}\_loss\), which calculates the mean squared error. We tried several hyperparameters, such as \(\lambda\), with values ranging from 0.005 to 0.05. We saved the best models' weights to be imported to the new training model. We experimented with different architectures and numbers of layers with different filter sizes for the discriminator architecture and presented our final architecture. The discriminator PatchGAN comprises five layers, which have around 11,000,000 trainable weights. The model receives two inputs and concatenates them using the concatenate layer. Afterward, the concatenated input is processed through five Conv2D layers with different filter sizes (64, 128, 256, 512, and 1024), a stride value of 2, and the same padding scheme. An activation function called LeakyReLU [34] is applied to the output tensors with an alpha value of 0.25.