## 3 Results
The coordinate differences between the nodes indicate that the maximum discrepancy is 0.05, which can be attributed to the smoothing effect of neural network optimization. The cumulative curves of both methods confirm the overall grid adaptation trend, though the numerical method exhibits more pronounced slope changes. The loss function graph shows that the PINN method requires more iterations to achieve an accurate result but ultimately reaches high precision, minimizing the error to 0.000026. These results highlight the characteristics of each method and provide insights into their applicability for adaptive comp  
Figure 8: Graphs of node distribution at coordinates 0.3 and 0.7. Hard Enforcement of Boundary Conditions: Comparison of Approaches  
To evaluate the effectiveness of the hard enforcement of boundary conditions using the mask \(\phi(x)\), a series of numerical experiments are conducted comparing two approaches: implementing boundary conditions through an additional penalty term in the loss function and enforcing them explicitly by modifying the neural network output. Figure 9(a) shows the node distribution obtained using the mask \(\phi(x)\), while Figure 9(b) presents the distribution resulting from the loss-based boundary implementation. Visually, the overall structure of the mesh in both cases is similar, with the node density adapted in the central region of the domain according to the control function. However, the exact boundary values reveal a key difference between the methods. When using the mask, the coordinates at the interval boundaries match the target values precisely: \(s(0)=0.00000000\), \(s(0)=1.0000000\). In contrast, when boundary conditions are enforced via the loss function, noticeable deviations are observed: \(s(0)=0.00554807\), \(s(1)=0.97303724\). Figure 10: Node distribution with (**a**) hard enforcement of boundary conditions and (**b**) loss-based enforcement. Figure 9: Graphs of node distribution at coordinates 0.3 and 0.7. These results demonstrate that the use of the mask \(\phi(x)\) enables the strict enforcement of Dirichlet boundary conditions with machine precision, without the need to manually tune the penalty weights in the loss function. This is particularly important in problems requiring the exact placement of boundary nodes, such as in simulations with a prescribed geometry or fixed boundary values. In the future, additional metrics will be used for a more detailed analysis of the differences between the numerical method and the PINN method. Specifically, quantitative grid deviation measures will be considered, such as the root mean square error between node coordinates, a smoothness measure of the node distribution, and PINN convergence indicators depending on various loss function parameters. This will allow for a more objective assessment of the accuracy and efficiency of each approach and help identify potential improvements in adaptive computational grid generation. This section presents an analysis of the results for constructing a one-dimensional computational grid using both the numerical solution method and the PINN method. Cases with different numbers of control points defining node clustering zones are examined. An analysis of the node distribution, step size between neighboring points, coordinate differences, cumulative curves, and loss function shows that the PINN method produces smoother grids than the numerical method. As the number of control points increases, the differences between the methods become more pronounced. It is also observed that the convergence rate of the PINN method varies depending on the complexity of the problem, but the model successfully adapts to parameter changes. The obtained results provide insights into the applicability of PINNs for adaptive computational grid generation in numerical modeling tasks. However, it should be noted that the numerical results used for comparison are not an absolute benchmark.