## 3 Methodology  
### Tabular Generative Adversarial Networks  
To address the scarcity of available data, GANs have been proposed as a solution for generating synthetic data from original data [32]. GANs draw inspiration from the concept of zero-sum games in game theory. In this framework, there exist two networks: a generator network (G) and a discriminator network (D). The GAN leverages the independent and adversarial relationship between G and D to train and produce specified data samples. G receives a random noise vector as input, from which it generates new relevant data. D, on the other hand, receives both the actual data and the data generated by G as input. These two networks engage in continuous competition, each pursuing its own objectives  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Refs.** & **ML Technology** & **Dataset** & **Inputs** \\ \hline
[29] & DT, Bagging, LightGBM & 351 & Specim age, molarity of NaOH, natural zeolite, silica fume, GGBFS \\ \hline
[22] & ANFIS, ANNs, GEP & 245 & Chemical composition of fly ash, mixing procedures, curing regime, activator content, fine aggregate, coarse aggregate, water, activator/fly ash, and molarity of NaOH \\ \hline \hline \end{tabular}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{c}{} & LR: Linear Regression, EL: Elastic Net, SVM: Support Vector Machine, GPR: Gaussian Process Regression, DT: Decision Tree, AdaBoost: Adaptive Boosting, RF: Random Forest, GB: Gradient Boosting, lightGBM: Light Gradient Boosting Machine, XGB: XGBoost, kNN: k-Nearest Neighbors, MVR: Multivariate Regression, CatB: CatBoost, ANN: Artificial Neural Network, MLR: Multinomial Logistic Regression, MNLR: Multinomial Naive Bayes Logistic Regression, MSP: M5’ Model Tree, MARS: Multivariate Adaptive Regression Splines, GA: Genetic Algorithm, PSO: Particle Swarm Optimization, GWO: Grey Wolf Optimizer, BLR: Bayesian Linear Regression, Bagging: Bootstrap Aggregating, LSBoost: Least Squares Boosting, ANFIS: Adaptive Neuro-Fuzzy Inference System, GEP: Gene Expression Programming. \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._independently. As the distribution of the generated data gradually converges to that of the real data, it becomes increasingly challenging for D to differentiate between them, signaling the completion of the GAN model training. Remarkably, various GAN variations have emerged for generating synthetic data, spanning images, text, and numerical data, among others. One notable advancement is the TGAN, which excels in generating both discrete and continuous tabular data simultaneously, showcasing promising efficiency [12].  
TGANs employ the Adam optimizer for training, with a long short-term memory (LSTM) network acting as the generator and a multi-layer perceptron (MLP) functioning as the discriminator. One interesting feature of a TGAN is its capability to transform non-Gaussian distributions into continuous columns using a clever mode-specific normalization method. This enables it to capture the inherent distribution of the original data and effectively model the relationships between features. In the optimization phase, the loss function incorporates both KL divergence and cluster vector terms to enhance model stability. The formulation of the loss function is provided in Equations (1) and (2).  
Generator:  
\[Loss_{G}=-E_{z\;N(0,1)}logD(G(z))+\sum_{i=1}^{n_{c}}KL(u_{i}^{\prime},u_{i})+ \sum_{i=1}^{n_{c}}KL(d_{i}^{\prime},d_{i}) \tag{1}\]  
Discriminator:  
\[Loss_{D}=-E_{v_{1:n_{c}},u_{1:n_{c}},d_{1:n_{d}}}\;p_{(T)}logD(v_{1:n_{c},u1:n _{c},d1:n_{d}})+E_{z\;N(0,1)}logD(G(z)) \tag{2}\]  
where \(u_{i}^{\prime}\) and \(d_{i}^{\prime}\) are synthetic data, \(u_{i}\) and \(d_{i}\) are original data, and \(n_{c}\) and \(n_{d}\) are the numbers of continuous and discrete variables, respectively.  
Figure 1 illustrates the architecture of a TGAN. In this investigation, a TGAN was trained utilizing the TGAN package implemented in Python.