# Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy
## 1 Introduction  
Physics-Informed Neural Networks (PINNs) are a rapidly emerging numerical technique used to solve partial Differential equations (PDEs) by means of a deep neural network. The first idea can be traced back to the works of [PERSON] et al. [1; 2; 3], but, thanks to the hardware advancements and the existence of deep learning packages like Tensorflow [4], Pytorch [5] and JAX [6], they have recently became popular since the works of [PERSON] et al. [7; 8], published in [9]. In its original formulation, the approximate solution is computed as the output of a neural network trained to minimize the PDE residual on a set of collocation points inside the domain and on its boundary. The growing interest in PINNs is strictly related to their flexibility. In fact, with minor changes to the implementation, it is possible to solve a huge variety of problems. For example, exploiting the nonlinear nature of the involved neural network, nonlinear [10; 11] and high-dimensional PDEs [12] can be solved without the need for globalization methods or additional nonlinear solvers. Moreover, by changing the neural network's input dimensions or suitably adapting the loss function, it is possible to solve parametric [13; 14] or inverse [15; 16] problems. When external data are available, they can also be used to guide the optimization phase and improve the PINN accuracy [17]. In order to improve the original PINN proposed in [9] and to adapt it to solve specific problems, several generalizations have been proposed. For example, the deep Ritz method (DRM) [18; 19; 20] looks for a minimizer of the PDE energy functional and, in the deep Galerkin method (DGM) [21; 22; 23], an approximation of the \(L^{2}\) norm of the PDE residual is minimized. It is also possible to exploit domain decomposition strategies [24; 25] as in the conservative PINN (CPINN) [26], in the parallel PINN [27], in the extended PINN (XPINN) [28], or in the Finite Basis PINN (FBPINN) [29]. Moreover, it is even possible to change the neural network architecture or the training strategy as in [30; 31; 32; 33; 14; 34; 35]; between the methods based ondifferent architectures, we highlight some works based on the novel Kolmogorov-Arnold Network (KAN) [36] architecture [37; 38] and on a Large Language Model (LLM) [39]. More extensive overviews of the existing approaches can be found in [40; 41; 42; 43]. In the context of the current work, an important extension is the Variational-Physics-Informed Neural Network (VPINN) [44; 45], where the weak formulation of the problem is used to construct the loss function. In this work, we focus on VPINNs. As discussed in [44; 45; 46; 47], in order to train a VPINN, one needs to choose a suitable space of test functions, compute the variational residuals against all the test functions on the basis of such a space, and minimize a linear combination of these residuals. Since a spatial mesh is required to define the test functions, the VPINN cannot be considered a meshfree method, even though it is an extension of the PINN, which is meshfree. In this work, we present an adaptive Meshfree VPINN (MF-VPINN) that does not require a global triangulation of the domain but is trained with the same loss function and neural network architecture of a standard VPINN. Note that the MF-VPINN and the original VPINN can solve the same differential problems because the neural network is trained with the same loss functions. We also highlight that they can solve problems where the solution has low regularity that cannot be solved with standard PINNs, for example, in the presence of singular forcing terms, thanks to the weak formulation of the PDE without introducing further approximations or regularizations. However, one of the VPINN's limitations is that a triangulation of the entire domain is required to define the test functions.