## 1 Introduction  
In the last decade, the widespread use of artificial neural networks (ANNs) led to a significant step forward in computer science and computational mathematics. A wide range of practical problems, including segmentation and classification tasks, could be efficiently solved using this tool. Here, the related mathematical models fall into the class of discrete problems. At the same time, we have to keep in mind that the engine of the ANN-based algorithms is a gradient-based continuous optimization algorithm enriched with stochastic elements. This gives the first motivation to also apply ANN-based techniques for the numerical solution of problems arising from continuous mathematical models. The numerical solution of partial differential equations (PDEs) constitutes an important class of these, having a central role in the natural sciences and engineering. Accordingly, a family of ANN-based methodologies was developed for solving partial differential equations (PDEs) numerically. The most common approach is the class of the so-called physics-informed neural networks (PINNs). In this framework, the solution candidates are optimized based on the most straightforward error terms. To compute these, one has to calculate consistency error in the governing equations and boundary conditions. In concrete terms, the approximations are substituted into the physical laws constituting the mathematical problem. For a general description of this method, we refer to [1, 2]. A number of specific forms of this method were developed and applied for a scale of classical PDE problems arising in real-life applications; see, e.g., [3]. Beyond these, the above framework can be extended to the numerical solution of stochastic [4] and inverse problems [5, 6] as well. A possible shortcut of this procedure is to seek approximations only from a class of functions that satisfy the governing equation exactly. In this case, we only have to care about parameters (unknowns) associated with the boundary, which can largely decrease the dimension of the corresponding optimization problems. Such an algorithm is introduced in [7] and applied to the numerical solution of Dirichlet-to-Neumann problems. To summarize these developments and further approaches in detail, a number of review articles have been published in the past years. Among them, we propose [8; 9], where a really wide range of further specific references can be found. All of these approaches offer a complete procedure for the numerical solution of PDEs. At the same time, in the last 75 years, a wide class of conventional numerical methods were developed for solving PDEs. A natural idea is to combine these with the efficient computational tools offered by the ANNs. This is also the aim of the present study. In other words, we propose here an _ANN-assisted_ numerical solution of PDEs. In this framework, instead of the entire approximation procedure, only an important and technical step of a conventional numerical method is executed using an ANN. In the course of the numerical solution of a PDE, we usually first have to discretize the underlying problem including the governing equation and the boundary data. Whenever we can choose purely function-space-based discretizations, it is mostly connected to a physical discretization of the computational domain, where the equation is posed. Generating a suitable mesh for this purpose is still a challenging problem. At this step, one can also make use of ANN-based procedures; see, e.g., [10; 11; 12]. Having this mesh at hand, choosing a specific finite element or finite difference discretization method and, finally, incorporating the given boundary data, we obtain an algebraic (or linear algebraic) system of equations. The construction of such finite-difference-based approximations will be the focus of our study. In case of uniform rectangular meshes, we can easily derive finite difference approximations of various differential operators and convert the original problem quickly into a system of equations.