## 1 Introduction  
Using artificial neural networks (ANNs), a number of computationally complex or data-intensive computational problems have been successfully tackled in the last decade. Furthermore, ANNs can be applied to computational tasks with no rigorous mathematical model in the background; the structure of the neural network itself serves as a (sometimes, heuristic) model. A natural idea is to employ this powerful tool for the classical computational problems such as the numerical solution of partial differential equations (PDEs). Accordingly, a number of different algorithms have been proposed. The most-widespread family of the different approaches is the so-called physics-informed neural networks (PINNs), which were introduced in [1]. For a detailed review on this topic, we refer to [2]. In the first version of this popular family of methods, called the data-driven PINNs, we have a fixed PDE \(\partial_{t}u(t,\mathbf{x})+D_{\mathbf{x}}u(t,\mathbf{x})=0\) for the unknown function \(u:(0,T)\times\Omega\to\mathbb{R}\), where \(\Omega\) denotes the spatial computational domain and \(D_{\mathbf{x}}\) is a generic spatial differential operator. Furthermore, the equation should be equipped with an appropriate initial condition \(u_{0}\) and boundary data \(u_{\partial\Omega}\) for the well-posedness. To prepare the computations with the ANN, a huge number of solution candidates \(v_{j}\) should be computed such that these determine a large variety of initial and boundary conditions. If \(v_{j}\approx u\) is only an approximation, the right-hand side defined by \(f_{j}:=\partial_{t}v_{j}+D_{x}v_{j}\) is not necessarily zero. Furthermore, \(v_{j}\) has some initial value \(v_{j,\partial}\) and boundary value \(v_{j,\partial\Omega}\). Within the frame of the conventional PINNs, for a given set \(\{\mathbf{x}_{j}\}_{j=1}^{M}\subset\Omega\), an ANN is trained using the pairs:  
\[\Big{\{}([f_{j},v_{j,\partial},v_{j,\partial\Omega}],[v_{j}(t_{1},\mathbf{x} _{1}),\ldots,v_{j}(t_{M},\mathbf{x}_{M})])\Big{\}}. \tag{1}\]In this way, the ANN will try to learn the solution at some points using the left-hand side \(f_{j}\), the boundary data \(v_{j,\partial\Omega}\), and the initial data \(v_{j,0}\), such that it can be recognized as a solution operator. For a given structure of the ANN, the values of its parameters will be determined such that the error:  
\[[v_{j}(t_{1},\mathbf{x}_{1}),\ldots,v_{j}(t_{M},\mathbf{x}_{M})]-[\hat{v}_{j}(t _{1},\mathbf{x}_{1}),\ldots,\hat{v}_{j}(t_{M},\mathbf{x}_{M})]\]  
is minimal. Here, \(\hat{v}_{j}\) denotes the ANN estimate of the original function \(v_{j}\). To perform this, we should specify a norm to measure the error above. This norm is commonly called the loss function. Finally, applying this network to the input \((0,u_{0},u_{\partial\Omega})\), we obtain the desired approximation of \(u\). This original idea was developed and extended to incorporate unknown parameters in the equation. In this way, it is perfectly fit to solve inverse problems; see, e.g., [3]. The real-life case of noisy measurements or observation data can also be fit into this framework, linking this computational tool with the powerful Bayesian inversion approach [4]. Furthermore, the methodology of computing with PINNs was refined and discussed for a number of cases concerning the Navier-Stokes equations [5], as well as stochastic PDEs [6]. Beyond these specific applications, the PINN-based methods were also extended to make them flexible in the choice of the computational domain [7]. Due to the practical importance of developing efficient PDE solvers, a number of further ANN-based approaches have been proposed for this purpose. A new, promising methodology is based on mappings between function spaces. As the Galerkin methods became the leading tool in the case of conventional numerical methods, the function space approach, instead of working pointwise or with mesh-dependent data, seems to be an adequate direction for the learning algorithms.