Застосування методів машинного навчання для апроксимації розв'язків диференціальних рівнянь

Abstract

This paper explores the application of machine learning methods for approximating solutions of differential equations. The effectiveness of neural networks such as Physics-Informed Neural Networks (PINNs), Deep Neural Networks (DNN), and Convolutional Neural Networks (CNN) is analyzed in solving ordinary differential equations and partial differential equations. The study evaluates the ability of machine learning to adaptively approximate complex functions, their accuracy, performance, and capability to handle high-dimensional problems. Special attention is given to automatic differentiation, loss function optimization, and numerical solution generation. The results demonstrate the practicality of using ML methods in scientific computing, engineering simulations, and mathematical modeling.