## 4 The Proposed Model
### Objective Function  
GANs are unsupervised models based on game theory, proposed by [PERSON] et al. in 2014. In this model, the generator \(\mathfrak{S}\) maps random noise onto the target domain with a particular distribution (such as Gaussian or uniform), and the probability distribution of the real data is then studied in order to generate samples \(\mathfrak{S}(z)\) that closely match the real data distribution. To avoid the mode collapse problem when random noise is used, \(\mathfrak{D}\) identifies whether the sample derives from the real dataset \(x\) or from the generated dataset \(\mathfrak{S}(z)\). \(\mathfrak{D}(\cdot)\) is a probability value that implies whether the dataset is real. Training a GAN primarily involves training the discriminator \(\mathfrak{D}\) and generator \(\mathfrak{S}\), where \(\mathfrak{S}(z)\) is trained to reduce the likelihood of \(\mathfrak{D}\) making mistakes, whereas \(\mathfrak{D}\) is used to increase the precision \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{R_{T}}\) & **Mw** & **MF** & **Area\%** & **Compound Name** \\ \hline
11.40 & 136 & C10H16 & 0.26 & Bicyclo-[3.1.1]-hept-2-ene,3,6,6-trimethyl (CAS) \\ \hline
11.71 & 136 & C10H16 & 0.61 & 1,3,6-Octatriene,3,7-dimethyl-(-(E)-(CAS) \\ \hline
12.00 & 136 & C10H16 & 0.90 & \(\mathfrak{S}\)-Terpinene \\ \hline
12.88 & 136 & C10H16 & 0.90 & \(\mathfrak{S}\)-Terpinene \\ \hline
13.41 & 154 & C10H18O & 9.88 & Linalool \\ \hline
14.26 & 218 & C\({}_{9}\)H\({}_{15}\)BrO & 0.73 & 1a-Bromo-3A\({}_{4}\),3a,5,6,7,7A\({}_{\text{A}}\)-Hexahydroindan-4-ol \\ \hline
17.30 & 154 & C9H14O2 & 1.40 & MethylBicyclo [3.1.0]hexane-6-acetate \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._level for distinguishing real from generated samples [17]. Hence, training \(\mathfrak{S}(z)\) and \(\mathfrak{D}\) is a binary minimax game problem that is characterized by the following objective function:  
\[J(\mathfrak{D},\mathfrak{S})=\mathbb{E}_{x\sim p_{data}}[\log\mathfrak{D}(x)]+ \mathbb{E}_{z\sim p_{z}}[\log(1-\mathfrak{D}(\mathfrak{S}(z)))] \tag{1}\]  
where \(x\sim p_{data}\) implies that datum \(x\) follows some \(p\) data distribution, and \(z\sim p_{z}\) implies that the random noise \(z\) follows the distribution \(p_{z}\). When the generator trains a GAN, it attempts to generate progressively realistic images so that a discriminator will not be able to distinguish between real and false images, whereas \(\mathfrak{D}\) attempts to distinguish between real images and those generated by \(\mathfrak{S}\). This is similar to a two-player game in which the generator and the discriminator are steadily at odds. Eventually, a dynamic equilibrium is reached between the two networks; the image generated by \(\mathfrak{S}\) represents the real image, whereas the discriminator cannot distinguish between them. As a result, network training is ultimately intended to maximize the probability values of the \(\mathfrak{S}\) and \(\mathfrak{D}\) networks, which means that \(\mathfrak{S}\) and \(\mathfrak{D}\) should achieve \"Nash equilibrium\". In order to stabilize the training of GANs, different objective functions have been proposed, including f-divergence [41], least squares [42], Wasserstein distance [43], and hinge loss [44]. The Wasserstein metric is feasibly the most favored for the measurement of distances between real and generated samples, developing a new GAN framework called the Wasserstein GAN (WGAN) [43]. A significant advantage of the WGAN over the original GAN is its improved theoretical properties and learning stability, which facilitate hyperparameter selection and debugging. [PERSON] et al. [45] improved the original WGAN by penalizing the norm of the discriminator gradients with respect to the data input rather than clipping the weight parameters of the discriminator. It is often possible to parameterize networks \(\mathfrak{D}\) and \(\mathrm{G}\) according to differentiable functions that may include fully connected, convolutional, or recurrent networks.