## 1 Introduction
Thus, the application of differential equations in adaptive mesh generation is not merely a technical tool but rather a fundamental mathematical framework for creating unified and controllable models of computational meshes adapted to the physics of the problem and the geometry of the domain. These approaches have proven effective in computational fluid dynamics and finite element analysis, ensuring high mesh quality and alignment with the physical features of the solution [11]. One-dimensional computational grids are widely used in various applied problems, such as hydrodynamics, continuum mechanics, and electrodynamics. This study examines the construction of a one-dimensional adaptive computational grid based on the diffusion equation. This equation regulates grid density by refining the node distribution in regions with steep solution gradients. Existing research has employed different approaches to construct adaptive computational grids using the diffusion equation. The study in [12] presented an unstructured grid generation method using differential techniques, achieving adaptation via the inverse Beltrami equation. The authors proposed using Delaunay triangulation for final grid construction, ensuring the smoothness of the geometric characteristics of grid cells. In [13], the application of Beltrami and diffusion equations for numerical grid generation was explored, taking into account the physical properties of the modeled medium. Later, [14] introduced an algorithm for generating three-dimensional adaptive grids, utilizing inverse Beltrami equations to construct grids in multiscale domains. The study in [15] examined variational and elliptic grid generation methods that optimize node distribution while considering boundary conditions. Additionally, [16] proposed a fragmented algorithm for adaptive grids, implementing automated computational fragment management in the LuNA system, which efficiently distributes computing resources for complex simulations. In recent years, PINNs have gained significant attention for solving differential equations, including the diffusion equation. Numerous studies have demonstrated the successful application of PINNs to problems involving heat conduction, mass transport, and other processes described by diffusion-type models [17; 18; 19]. By embedding physical laws directly into the loss function, such models can produce stable and interpretable solutions, even when data are limited or noisy. The studies in [20; 21; 22; 23] explore various extensions of PINNs, including hybrid models, variational approaches (VPINN), and Bayesian methods (B-PINN), which improve solution accuracy and stability. In these methods, the network is trained with physical constraints, reducing dependence on large training datasets. The study in [24] investigates the application of PINNs to solve equidistribution equations for constructing two-dimensional adaptive computational grids. PINNs allows for approximate equation solutions without explicit discretization by integrating physical laws into the loss function. Recent research has utilized neural networks for automated grid generation and adaptation. The study in [25] applies graph neural networks (GNNs) to predict and optimize grids in hydrodynamics and mechanics. In [26], a method for generating finite element meshes based on self-organizing Kohonen maps is proposed. The study in [27] explores the use of GNNs for adaptive grid generation in numerical methods. These studies confirm the potential of machine learning in optimizing grid construction, reducing computational costs, and enhancing simulation accuracy. However, the use of PINNs specifically for adaptive mesh generation through node distribution control based on the diffusion equation has received little attention in the literature. Existing works on PINNs typically focus on the accuracy of the solution but do not examine the structure of the computational mesh implicitly formed during training.