Adaptive Grid Generation by Solving the Two-Dimensional Diffusion Equation Using Physics-Informed Neural Networks

Adaptive computational grids improve the accuracy and efficiency of numerical simulations by concentrating nodes in regions of particular interest, but classical adaptation methods based on differential equations are often computationally expensive and geometry-dependent. This study proposes a physics-informed neural network (PINN) framework for generating two-dimensional structured adaptive grids by solving the diffusion equation directly, with grid density governed by a prescribed control function. Two neural architectures were developed and compared: a classical PINN, in which boundary conditions are enforced softly through an additional loss term, and a modified PINN (MPINN), in which Dirichlet boundary conditions are embedded exactly into the network architecture using an approximate distance function (ADF) mask. Both methods were tested on two adaptation scenarios—concentration at a single point and concentration along a line segment—and validated against a reference grid obtained by a finite-difference scheme. The MPINN model reduced the mean root-mean-square error by a factor of between 1.4 and 2.0, depending on the adaptation case, together with improved boundary agreement, symmetry, and interval uniformity. Two further configurations with control functions of a more complex shape, a T-shaped junction and a curve of diagonal sine form, were examined together with a variant in which the boundary nodes are allowed to slide along the boundary while the boundary itself is preserved exactly. In these configurations the classical PINN fails to preserve the domain, whereas both mask-based variants reproduce it by construction. The results demonstrate that embedding boundary conditions architecturally, rather than penalizing their violation in the loss function, substantially improves the accuracy and reliability of PINN-based adaptive grid generation.