# Dynamic Simulation Model-Driven Fault Diagnosis Method for Bearing under Missing Fault-Type Samples
## 2 Theoretical Background
### Wasserstein Generative Adversarial Networks with Gradient Normalization (WGAN-GN)  
The Generator (G) and Discriminator (D) are the two components that make up the GAN structure, as shown in Figure 4. A random noise vector is utilized as the input of the G, which attempts to create realistic data to deceive D. The D then learns to distinguish between actual data and G-produced synthetic data. Typically, G and D are constantly \begin{table}
\begin{tabular}{c c} \hline \hline
**Description of Parameters** & **Values of Parameters** \\ \hline The radius of outer race/mm & 17.0 \\ The radius of inner race/mm & 39.9 \\ Pitch diameter/mm & 28.3 \\ Diameter of rolling element/mm & 6.8 \\ Number of balls & 8 \\ Contact angle/\({}^{\circ}\) & 0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of 6203. Figure 3: Time\(-\)domain waveforms of simulation data. Figure 2: Schematic of the local defect area: (**a**) Outer race fault, (**b**) Inner race fault, (**c**) Ball fault. optimized to enhance their ability to generate and discriminate data, and they are often parameterized as deep neural networks and optimize a mini-max objective. The training objective function of the GAN is displayed as follows:  
\[\underset{G}{\text{minimax}}\mathbb{E}_{x\sim p_{r}(x)}[\log(D(x))]+\mathbb{E}_{ \widetilde{x}\sim p_{g}(x)}\Big{[}log\Big{(}1-D\Big{(}\widetilde{x}\Big{)} \Big{)}\Big{]} \tag{9}\]  
where \(p_{r}(x)\) is the distribution of raw data and \(p_{g}(x)\) is the distribution defined by \(p_{g}=G_{*}(p_{z})\), * is the push-forward measure and \(p_{z}\) is the distribution of stochastic noise vector. If D is always optimal in this scenario, the G will inevitably converge to the true distribution \(p_{r}(x)\). However, gradient vanishing and gradient explosion are the two fundamental problems that training GAN always faces. Optimizing Equation (1) is comparable to minimizing the Jensen-Shannon divergence (JS-divergence) between \(p_{g}(x)\) and \(p_{r}(x)\) for these unstable training situations. If \(p_{g}(x)\) and \(p_{r}(x)\) do not overlap, the gradient would vanish since the JS-divergence would be a constant number. Second, D frequently succumbs to overfitting with finite real samples, which subsequently results in gradient explosion near real samples. So the JS-divergence may be not a suitable cost function. A series of recent studies have focused on solving the problem of unstable training. The steep gradient space of the discriminator is one of the causes of unstable GAN training, which can cause a pattern crash during the training of the generator. Although simple methods such as L2 normalization and weight clipping can effectively make the GAN training process more stable, the discriminator model capacity is constrained by these extra restrictions. So instead of learning to produce true data, the generator is more likely to trick the discriminator. The regularization or normalizing of the discriminator is another well-liked method for formalizing the discriminator as a Lipschitz continuous function. This way allows for the smoothing of the discriminator gradient space without compromising discriminator speed. The purpose of WGAN is to minimizing the Wasserstein distance between \(p_{g}(x)\) and \(p_{r}(x)\), i.e.,  
\[\underset{G}{\text{minimax}}\underset{D,L_{D}\leq 1}{\text{max}}\mathbb{E}_{x \sim p_{r}(x)}[D(x)]-\mathbb{E}_{\widetilde{x}\sim p_{g}(x)}\Big{[}D\Big{(} \widetilde{x}\Big{)}\Big{]} \tag{10}\]  
where \(L_{D}\) is the Lipschitz constant of discriminator \(D\). \(L_{D}\) is defined as follows:  
\[L_{D}\text{inf}:=L\in\mathbb{R}:]D(x)-D(y)\leq L\parallel x-y\parallel, \forall x,y\in\mathbb{R}^{n} \tag{11}\]  
Figure 4: Generative adversarial networks. where \(\|\cdot\|\) can be the norm of vector. By maximizing Equation (2) under the Lipschitz constraint in Equation (3), the discriminator in WGAN seeks to approach the Wasserstein distance.