APPLICATION OF PINN AND THE METHOD OF DIFFERENTIAL CONSTRUCTION OF ADAPTIVE ONE-DIMENSIONAL COMPUTATIONAL GRIDS

Abstract

This paper presents an approach to constructing adaptive one-dimensional computational grids using the Beltrami equation and Physics-Informed Neural Networks (PINNs). The main focus is on exploring the potential for precise control of grid node density through the control function ω(s), which allows the grid to adapt to the local features of the problem. The Beltrami equation used, being a key component of the method, regulates the distribution of nodes by modifying the function’s derivatives depending on the values of the control function. The effectiveness of this approach is demonstrated through examples involving one and two regions of node clustering.

The results showed that the PINN method combined with the Beltrami equation allows for the creation of computational grids with a high degree of adaptation to given conditions, providing detailed modeling in critical regions. This approach has advantages over traditional numerical methods, as integrating physical laws in the grid construction process minimizes numerical errors and improves modeling accuracy. The use of neural networks offers flexibility in model tuning and the ability to account for complex nonlinear dependencies. The discussion of the results highlights the potential of using PINNs for adaptive grid construction in various fields requiring precise and efficient modeling. In conclusion, this study confirms that the combination of the Beltrami equation and PINNs is a powerful tool for adaptive grid construction, opening new possibilities for numerical modeling of complex physical processes.