## 3 Results  
This section presents numerical experiments conducted to solve the one-dimensional diffusion equation given in the form (1). The control function plays a key role in forming the adaptive grid, allowing for node clustering in selected regions of the computational domain. In the numerical tests, the following form of the control function is used:  
\[\omega(s)=1+\sum_{i}\frac{\alpha}{\left(x-x_{i}\right)^{2}+\epsilon} \tag{13}\]  
where \(x_{i}\) are the coordinates of the points near which an increased node density is required, \(\alpha\) is the coefficient controlling the intensity of local refinement, and \(\epsilon\) is a smoothing parameter that prevents singularities and abrupt changes in density. During testing, the positions of \(x_{i}\), as well as the values of \(\alpha\) and \(\epsilon\), are varied to examine how different characteristics of the control function influence grid generation using both the numerical method and the PINN approach. To evaluate the characteristics of one-dimensional computational grid generation using both the numerical method and the PINN method, tests are conducted with various control function parameters. During the experiments, the parameters \(\alpha\) (which determines the intensity of node clustering) and \(\epsilon\) (which controls the degree of smoothing of the control function) are varied. A fixed control point at \(x=0.5\) is used, and the grid coordinates are distributed within the interval \([0,1]\). The tests utilize the following parameter combinations: \(\alpha=~{}0.01,~{}0.015,~{}0.02\) and \(\epsilon=~{}0.01,~{}0.02\). This allows for an evaluation of the impact of each parameter on grid adaptation and differences in node distribution. The test results are presented as graphs analyzing the node distributions for both methods, spacing between points, coordinate differences between nodes, cumulative curves, and convergence rate of the loss function. Below is a detailed analysis of each of these aspects. Figure 3 illustrates the impact of parameter \(\alpha\) on the node distribution in both the numerical method and the PINN method. As \(\alpha\) increases, local node clustering intensifies, which is more pronounced in the numerical method, where the node density near the control points increases significantly. In the PINN method, increasing \(\alpha\) also affects grid adaptation; however, clustering occurs more smoothly, without abrupt jumps in the distribution. Parameter \(\epsilon\) determines the degree of smoothing of the control function, and its increase leads to a more uniform node distribution in both methods. In the numerical method, this results in reduced sharp step variations, while in the PINNs, smoothing causes less pronounced node density adaptation around the control point, making the distribution more uniform. In the numerical method, more pronounced step size minima are observed near the control points, corresponding to the given function \(\omega(s)\). In the PINN method, step size variations occur more smoothly, without abrupt minima, which may be related to the optimization characteristics of the model. The shapes of the graphs in both methods maintain the general trend of step size reduction in the regions of node clustering and increase near the boundaries of the computational domain. However, in the numerical method, the step changes are more abrupt, whereas in the PINNs, the transitions are smoother. Figure 5 shows a graph of the absolute difference between the node coordinates in the two methods. It can be seen that the largest discrepancies occur near the region of node clustering. For small values of \(\alpha\), the difference between the methods remains insignificant. However, as \(\alpha\) increases, the numerical method exhibits a sharper change in grid density, leading to greater coordinate deviations. Parameter \(\epsilon\) also affects the nature of the differences between the methods: as \(\epsilon\) increases, the discrepancies decrease since both methods exhibit less pronounced grid clustering.