## 5 Numerical Examples  
### Numerical Results of Curvature Interference Limit Lines  
The basic parameters of the face worm gear drive can be determined based on the gear manual [32], as summarized in Table 1. The process parameters of the worm are shown in Table 2.  
The existence of two contact lines was first determined using the geometric construction method, and the procedure is described in ref. [26]. To calculate the curvature interference limit line, five key points are uniformly selected within the coordinate system \(\left(O_{1};\sqrt{x_{2}^{2}+y_{2}^{2}},y_{\mathrm{\sigma}1}^{*}\right)\), with their coordinate values listed in Table 3. The variable \(L\) values at these five key points are then substituted into the function \(\psi(\theta)\) in Equation (19).  
The neural network architecture consists of 20 hidden layers, each containing 26 neurons. For the L-BFGS optimization algorithm, the initial learning rate was set to 0.01, with a cosine annealing learning rate scheduler employing a minimum learning rate of 0.01. The training process was terminated once the loss function within a subdomain reached the desired accuracy. The loss function converged to the expected accuracy within 500 epochs and exhibited stable performance throughout the training, as illustrated in Figure 8. The training was conducted on an Intel(R) Core(TM) i7-8700 CPU with 16 GB of RAM, with a memory footprint of approximately 528.7 MB. The PINN model required approximately 1.95 s to reach the desired precision. In contrast, the Fsolve method achieved the same level of accuracy in approximately 0.02 s. While the Fsolve method remains an ideal method for solving nonlinear systems in traditional curvature interference boundary analyses, the PINN model is more suitable when inverse design parameter estimation is required.  
\begin{table}
\begin{tabular}{c c c} \hline
**Nomenclature (Unit)** & **Symbols/Formulas** & **Results** \\ \hline Center distance (mm) & \(a\) & 100 \\ Transmission ratio & \(i_{12}\) & 55 \\ Number of worm threads & \(Z_{1}\) & 1 \\ Number of worm gear teeth & \(Z_{2}=i_{12}Z_{1}\) & 55 \\ Radius of worm pitch circle (mm) & \(r_{1}\approx 0.68a^{0.85}/2\) & 20 \\ Modulus (mm) & \(m\approx 2a/i_{12}Z_{1}\) & 4 \\ Spiral parameters of worm (mm) & \(p=mZ_{1}/2\) & 2 \\ Worm axial mounting distance (mm) & \(z_{\mathrm{A}}=k_{12}a\), (\(k_{12}=0.6\)) & 60 \\ Worm’s actual face width (mm) & \(\mathrm{Lw}=0.73a\) & 73 \\ Radius of worm addendum circle (mm) & \(r_{\mathrm{a1}}=r_{1}+h_{\mathrm{a1}}^{*}m\), (\(h_{\mathrm{a1}}^{*}=1\)) & 34 \\ Radius of worm dependum circle (mm) & \(r_{\mathrm{f1}}=r_{1}-h_{\mathrm{f1}}^{*}m\), (\(h_{\mathrm{f1}}^{*}=1.2\)) & 26 \\ Tooth width of face worm gear (mm) & \(k_{\mathrm{g}}\) & 0.5 \\ \hline \end{tabular}
\end{table}
Table 1: Design parameters of face worm gear drive.  
\begin{table}
\begin{tabular}{c c c} \hline
**Nomenclature (Unit)** & **Symbols/Formulas** & **Results** \\ \hline Radius of grinding wheel (\({}^{\circ}\)) & \(r_{\mathrm{g}}\) & 150 \\ Half taper angle of grinding wheel for \(i\)-flank (\({}^{\circ}\)) & \(\delta_{\mathrm{g}}^{(1)}\) & 65 \\ Tilt angle of grinding wheel for \(i\)-flank (\({}^{\circ}\)) & \(\varepsilon_{1}\) & 5 \\ Half taper angle of grinding wheel for \(e\)-flank (\({}^{\circ}\)) & \(\delta_{\mathrm{g}}^{(2)}\) & 60 \\ Tilt angle of grinding wheel for \(e\)-flank (\({}^{\circ}\)) & \(\varepsilon_{2}\) & 5 \\ Operating center distance (mm) & \(a_{\mathrm{d}}=r_{\mathrm{f1}}+r_{\mathrm{g}}/\cos\varepsilon_{\mathrm{s}}(S=1,2)\) & 165.57 \\ \hline \end{tabular}
\end{table}
Table 2: Operating parameters for machining worm.  
Figure 8: Performance evaluation of the developed PINN model: (**a**) effect of the number of layers and neurons on the training time; (**b**) training loss curve.