# Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy
## 4 Conclusions and Discussion  
In this work, we presented a Meshfree Variational-Physics-Informed Neural Network (MF-VPINN). It is a PINN trained using the PDE variational formulation that does not require the generation of a global triangulation of the entire domain. In order to generate the test functions involved in the loss computation, we use an a posteriori error estimator based on the one discussed in [46]. Using such an error estimator, it is possible to add test functions only in regions in which the error is higher, thus increasing the efficiency of the method.  
We highlight that the main advantages of the method are that it is meshfree, as it requires only a covering of the domain with patches that can be of different shapes and that it automatically improves the solution with the application of local patches without requiring a global mesh manipulation. It can be therefore used in domains where it is expensive or impossible to generate a mesh. On the other hand, if a mesh suitable to describe the solution can be generated, a standard VPINN is preferable since the implementation is simpler and the convergence rate with respect to the number of test functions is higher.  
We discuss several strategies to generate the set of test functions. We observe that adding a few test functions inside the patches associated with higher errors while ensuring a smooth transition between regions with large patches and regions with small patches is the best way to obtain accurate solutions. We also show that, if the a posteriori error indicator is not used, the model's accuracy decreases and the training is slower.  
In this paper, we only focus on second-order elliptic problem even though VPINNs can be used to solve more complex problems. In a forthcoming paper, we will adapt  
Figure 18: Problem (35): Representation of the last set of patches obtained with the different strategies. Each dot represents a patch \(P_{i}\), its position is the center \(c_{P_{i}}\) of the patch, its size is proportional to the patch size \(h_{i}^{2}\), and its color is associated with the quantity \(r_{h,i}^{2}(u^{\mathcal{N}})\). The black rectangles represent the holes \(H_{i}\), \(i=1,2,3,4\). **(a)**_Strategy_\(\#1\), \(C_{M}=4\); **(b)**_Strategy_\(\#2\), \(C_{M}=4\); **(c)**_Strategy_\(\#1\), \(C_{M}=9\); **(d)**_Strategy_\(\#2\), \(C_{M}=9\).  
the a posteriori error estimator and analyze the MF-VPINN performance on other PDEs. Moreover, we are interested in the analysis of the approach in more complex domains (in which the patches have to be suitably deformed) and in high-dimensional problems, where using a standard VPINN is not practical.  
Conceptualization, S.B. and M.P.; methodology, S.B. and M.P.; software, M.P.; validation, M.P.; formal analysis, S.B. and M.P.; investigation, S.B. and M.P.; resources, S.B.; data curation, M.P.; writing--original draft preparation, M.P.; visualization, M.P.; supervision, S.B.; project administration, S.B.; funding acquisition, S.B. All authors have read and agreed to the published version of the manuscript.  
The author [PERSON] kindly acknowledges partial financial support provided by PRIN project \"Advanced polyhedral discretisations of heterogeneous PDEs for multiphysics problems\" (No. 20204 LN5N5_003) and by PNRR M4C2 project of CN00000013 National center for HPC, Big Data and Quantum Computing (HPC) (CUP: E13C22000990001). The author [PERSON] kindly acknowledges the financial support provided by the Politecnico di Torino where the research was carried out.  
The data are available upon reasonable request.  
The authors declare no conflicts of interest.