NEUROEVOLUTIONARY COLLOCATION METHOD FOR SOLVING DIFFERENTIAL EQUATIONS

Abstract

The importance of the development of numerical methods for solving differential equations is determined by their wide application in important fields of science and technology. The fact that many physical and engineering phenomena can be mathematically described by differential equations, but it is often difficult to find their analytical solutions. This makes numerical methods of approximate solution crucial. These methods are necessary for computer modeling and simulation of complex technical systems. Taking into account the wide range of types of differential equations, approximate methods become a universal tool, adapted to solve complex 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. Neural networks with the addition of physical information in the form of a complex loss function are an innovative approach that combines traditional methods of solving physical problems with modern techniques of deep learning. In this approach, a neural network, which is typically used to approximate functions, receives as input not only input data but also physical information about the system or process it is modeling. 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. The paper examines the possibility of applying genetic algorithms to adjust the hyperparameters of neural networks approximating an unknown function.