# Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy
## 3 Numerical Results
### Adaptive Training Strategies
Each dot represents a patch \(P_{i}\), its position is the center \(c_{\mathcal{P}_{i}}\) of the patch, its size is proportional to the patch size \(h_{i}^{2}\), and its color is associated with the quantity \(\eta_{i}^{\gamma}\). (**a**) Representation of \(\mathcal{P}_{3}\); (**b**) Representation of \(\mathcal{P}_{4}\); (**c**) Representation of \(\mathcal{P}_{5}\); (**d**) Representation of \(\mathcal{P}_{6}\); (**e**) Representation of \(\mathcal{P}_{7}\); (**f**) Representation of \(\mathcal{P}_{8}\)._  
cannot be removed when the centers are randomly chosen as in _Strategy_\(\#1\) because, in that case, \(\mathcal{P}_{m+1}\) would not be a cover of \(\Omega\) anymore. We also highlight that, removing the patches \(P_{i}\) such that \(\eta_{i}^{\gamma}\in\overline{\eta}_{m}\cup\overline{\eta}_{m,\ell}\) and choosing \(A_{\text{ratio}}=1\), it is possible to satisfy the inequality  
\[\sum_{P_{i}\in\mathcal{P}_{m+1}}|P_{i}|\leq C|\Omega|,\]  
for any \(m\in\mathbb{N}\) and with \(C>0\) independent of \(m\). Such a bound on the sum of the area of the patches is useful to ensure that there exists a number \(N_{\text{patch\_per\_point}}\) such that any point inside \(\Omega\) belongs to at most \(N_{\text{patch\_per\_point}}\) patches. This property is useful to derive global error indicators. We choose to maintain \(A_{\text{ratio}}=1.25\) to compare the numerical results with the ones obtained using the previous strategies and to consider overlapping patches. We train an MF-VPINN with \(C_{M}=4\) and \(C_{M}=9\) as in the previous tests. The corresponding error decays are shown in Figure 9. It can be observed that the error decreases in a smoother way and that, as in the previous tests, choosing \(C_{M}=4\) or \(C_{M}=9\) does not lead to significant differences in the error behavior. The patches used during the training are represented in Figures 10 and 11. We highlight that, when compared with the patch distributions in _Strategy_\(\#2\), there exist much more patches far from the origin, and, most importantly, the closer the center of a patch to the origin, the smaller its size. Even though the error decays with \(C_{M}=4\) and \(C_{M}=9\) are qualitatively similar, it should be noted that the patch distribution with \(C_{M}=9\) is more skewed. In fact. its patches can be clustered into two subgroups: the first one containing larger patches and covering most of the domain the second one containing only small patches with centers very close to the origin. A similar distribution is obtained with \(C_{M}=4\)  
In both cases, it can be observed that there are no large patches very close to small ones. This is in contrast with the distributions obtained in _Strategy_#2 and leads to more stable solvers. Indeed, even though the test functions are not related to a global triangulation on the entire domain \(\Omega\), the current loss function is very similar to the one used in a standard VPINN with a good-quality mesh, i.e., a mesh in which neighboring elements are similar in size and shape. On the other hand, in _Strategy_#2, there exist large patches that are very close to small ones; this is equivalent to training a VPINN on a very poor-quality mesh. Such meshes, in the context of FEM, are strictly related to convergence and accuracy issues.