## 3 Results
The maximum deviation, depending on the parameters of the control function, ranges from 0.01 to 0.1, and most tests show a difference within 0.03-0.05. Figure 6 presents a cumulative node curve, which illustrates the overall trend of the node distribution across the computational domain and shows how uniformly the node coordinate changes with respect to its index. In the numerical method, small segments with changes in the curve's slope are observed near the control points, corresponding to local grid refinement. In the PINN method, the cumulative curve changes more uniformly, without sharp bends, indicating a smoother variation in the node density. The overall shape of the curves in both methods remains similar; however, the numerical method exhibits more pronounced slope changes, while in the PINNs, the transitions are less abrupt. Figure 7 presents loss function graphs, illustrating the convergence dynamics of the PINN method depending on the control function parameters. Figure 5: Graphs of the difference between node coordinates. In all tests, the loss function decreases to 0.001; however, the convergence rate varies. Increasing \(\alpha\) and decreasing \(\epsilon\) require more iterations to reach the specified error threshold, which may be due to the model's more complex adaptation to sharp changes in the control function. In tests with \(\alpha=~{}0.01\) and \(\epsilon=~{}0.02\), the error threshold is reached more quickly, whereas for \(\alpha=~{}0.015\) and \(\epsilon=~{}0.01\), significantly more epochs are required for convergence. The results indicate that smoothing the control function positively affects the convergence speed of the PINN method, while increasing the intensity of grid clustering leads to longer training times. To analyze the adaptation of the computational grid to multiple control points, tests are conducted with parameters \(\alpha=~{}0.015\) and \(\epsilon=~{}0.001\). In this case, node clustering  
Figure 6: Graphs of cumulative node curves. Figure 7: Loss function graphs for the PINN method. is considered near two control points, \(x_{1}=0.3\) and \(x_{2}=0.7\), within the interval \([0,1]\). This allows for an evaluation of how effectively the numerical method and the PINN method reproduce the specified grid density distribution when multiple regions with high node concentrations are present. As part of the testing, graphs are generated for the node distribution, step sizes between neighboring points, coordinate differences between the methods, cumulative curves, and convergence rate of the loss function in the PINN method. Below is a detailed analysis of the obtained results, including a comparison of both methods based on key indicators such as the degree of grid clustering, uniformity of the node distribution, deviations between the numerical method and PINNs, and specific features of neural network model convergence. Figure 8 shows the node distribution in the numerical method and the PINN method for control points \(x_{1}=0.3\) and \(x_{2}=0.7\). In both methods, an increase in the node density is observed near the control points. In the numerical method, node clustering occurs more abruptly, whereas the PINN method exhibits a smoother grid density distribution without sharp jumps. Despite differences in approach, both methods successfully adapt the grid to the given parameters. However, the numerical method produces more pronounced gradients in the node distribution, while the PINN method shows some smoothing in high-density regions. Figure 9 presents the key grid characteristics, including the step size variation between neighboring nodes, coordinate differences between the methods, cumulative node curves, and loss function. Both approaches demonstrate the expected node clustering near the control points; however, the numerical method results in more abrupt density variations, whereas the PINN method produces a smoother distribution.