# Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy
## 3 Numerical Results
### The Importance of the Error Indicator
#### Strategy #4: Adaptive strategy without the error indicator  
Let us now analyze the consequences of choosing the position of the new patches without using the error indicator. To do so, we consider _Strategy_#1 but, instead of considering the new centers inside the patches \(P_{i}\) with the highest values of \(\eta_{i}^{7}\), we add them inside the patches with the highest values of \(\tau_{hid}^{2}(u^{\Lambda N})\). Using the equation residuals is a common choice in PINN adaptivity because the residuals describe how accurately the neural network satisfies the PDE at that point. The obtained error decay is shown in Figure 13. It can be seen that the accuracy is worse than the ones obtained with the other strategies and that the convergence rate with respect to the number of patches is lower. In such a figure, we also compare the MF-VPINN with a standard VPINN trained with test functions defined on Delaunay meshes. Note that, when _Strategy_#2 or _Strategy_#3 is adopted, the MF-VPINN is more accurate than a simple VPINN, even though its main advantage resides in being a meshfree method.  
We highlight that, due to the low regularity of the solution, the expected convergence rate with respect to the number of test functions of an FEM solution computed on uniform refinements is \(-1/3\). Note that the convergence rate of the proposed MF-VPINN method is still close to \(-1/3\), even though it is a meshfree method (see Table 1). For completeness, we also remark that, if an adaptive FEM is used, the rate of convergence depends on the FEM order.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(C_{M}\) & _Strategy_ \#1 & _Strategy_ \#2 & _Strategy_ \#3 & _Strategy_ \#4 & **Reference VPINN** \\ \hline
4 & \(-0.213\) & \(-0.295\) & \(-0.283\) & \(-0.105\) & \(-0.232\) \\
9 & \(-0.294\) & \(-0.376\) & \(-0.287\) & \(-0.182\) & \(-0.232\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rates of convergence with respect to the number of test functions.  
Figure 12: Top row: error indicator \(ES_{m}\) and rescaled \(H^{1}\) error \(c\|u-u^{\Lambda N}\|\). Bottom row: loss function. Left column: curves for the training with patches in \(\mathcal{P}_{6}\) shown in Figure 5f. Right column: curves for the training with patches in \(\mathcal{P}_{2}\) in Figure 11a. (**a**) \(ES_{6}\) and \(c\|u-u^{\Lambda N}\|\) for patches in Figure 5f; (**b**) \(ES_{2}\) and \(c\|u-u^{\Lambda N}\|\) for patches in Figure 11a; (**c**) Loss function for patches in Figure 5f; (**d**) Loss function for patches in Figure 11a.  
Figure 13: Comparison between the relative \(H^{1}\) errors obtained at the end of each training iteration with different strategies to choose the position of the new patches. (**a**) \(C_{M}=4\); (**b**) \(C_{M}=9\).  
Coherently with Figure 13, the best strategies are _Strategy_ #2 and _Strategy_ #3, whereas the worst one is _Strategy_ #4, which does not exploit the error indicator. The poor performance of _Strategy_ #4 can also be explained by analyzing the corresponding patch distribution. Such distribution is shown in Figure 14 for \(C_{M}=4\) and in Figure 15 for \(C_{M}=9\). These plots highlight that the patches do not accumulate near the origin because the residuals of the patches closer to it are not significantly higher than the other ones. For example, note the different colors in Figures 4 and 14, since in both cases, we randomly choose the position of \