## 1 Introduction  
In many areas of mathematical physics, solving complex physical problems often requires numerical approaches, especially when analytical solutions are unavailable or impractical. One of the foundational components of these numerical methods is the construction of computational grids, which play a key role in both the accuracy and efficiency of simulations. Fine grids offer more accurate results, particularly in regions where the solution changes rapidly, while coarser grids help reduce the computational cost in smoother regions. Adaptive mesh generation techniques aim to strike a balance between these two needs by adjusting the grid dynamically, refining it in critical areas and coarsening it where possible. This adaptability is especially valuable in simulations involving sharp gradients or localized phenomena. Recent advancements in machine learning, particularly physics-informed neural networks (PINNs), have opened new possibilities for solving partial differential equations while incorporating physical laws directly into the learning process. This study explores the application of PINNs to adaptive grid generation for the one-dimensional diffusion equation. By leveraging the equidistribution principle within a neural network frame work, we aim to generate grids that respond automatically to the behavior of the solution, providing an efficient and accurate numerical tool for further simulations. The study in [1] is dedicated to the numerical modeling of two-dimensional gas flows using adaptive curvilinear grids. The method of equidistribution is applied, allowing for automatic node refinement in regions with high flow gradients, thereby improving calculation accuracy. The studies in [2; 3] explore the theoretical foundations of grid generation, including solutions to partial differential equations and variational methods. The studies in [4; 5] propose methods for generating two- and three-dimensional grids by solving elliptic equations and optimizing node distribution. High-performance computing studies in [6] demonstrate the potential of parallel computations for accelerated grid generation. Computational grids are generally classified as either structured or unstructured. Structured grids utilize algebraic, differential, and variational methods to maintain a uniform node distribution while adapting to the problem's physical characteristics [7]. In unstructured grids, Delaunay triangulation and Voronoi diagrams are employed to achieve efficient and uniform grid generation for complex geometries [8; 9]. Elliptic methods and adaptive strategies also play a crucial role in optimizing node density based on computational conditions [7]. Both approaches are actively evolving and widely used in numerical modeling. However, the adaptive modification of grid density remains a challenging task that requires careful consideration of the system's physical properties. One of the most powerful approaches to generating such meshes is to formulate the problem as a boundary value problem for a system of differential equations. In particular, inverse diffusion equations, Poisson equations, and especially Beltrami equations have found wide application in this regard [10]. The use of differential equations in mesh generation differs from their classical application. In typical problems of mathematical physics, differential equations describe physical processes (such as heat conduction, diffusion, and wave propagation), whereas in mesh generation, they are used to define the transformation of coordinates from the computational domain to the physical domain. In this context, the variables in the equations are interpreted as geometric parameters of the mesh (e.g., coordinate functions), and their derivatives represent mesh distortion measures or node density.