NEURAL NETWORK METHODS FOR SOLVING ELASTICITY PROBLEMS

Abstract

The significance of the development of approximate methods in solving differential equations is determined by their wide application in key fields of science and technology. Since many physical and engineering phenomena can be mathematically described by differential equations, but finding their analytical solutions is often a difficult task, numerical methods for an approximate solution become critically important. These methods are necessary for computer modeling and simulation of complex technical systems. Given the variety of differential equations, approximate methods are a universal tool suitable for solving important problems in various fields, and allow better consideration of the requirements of modern computing technologies. The use of neural networks for the approximate solution of differential equations is a promising direction in the field of scientific modeling. An innovative approach is to include in the network physical information in the form of a complex loss function, which combines traditional methods of solving physical problems with advanced techniques of deep learning. In this approach, a neural network designed to approximate functions receives not only input data, but also physical information about the system or process it models. This physical information can be included as additional parameters, constraints, or equations. The complex loss function takes into account the quality of approximation by the neural network, as well as the physical principles of the problem. This allows neural networks to adapt to physical constraints and provides an approximate solution of problems, taking into account important aspects of the physical structure. Usually, simple architectures are used, for example, direct signal propagation networks with a small number of layers. This paper investigates the possibility of solving non-linear elasticity problems using a neural network approach.