Using NeuralPDE.jl to solve differential equations - Belicheva, Daria M.; Demidova, Ekaterina A.; Shtepa, Kristina A.; Gevorkyan, Migran N.; Korolkova, Anna V.; Kulyabov, Dmitry S. (Discrete and continuous models and applied computational science, 2025)
This paper describes the application of physics-informed neural network (PINN) for solving partial derivative equations. Physics Informed Neural Network is a type of deep learning that takes into account physical laws to solve physical equations more efficiently compared to classical methods. The solution of partial derivative equations (PDEs) is of most interest, since numerical methods and classical deep learning methods are inefficient and too difficult to tune in cases when the complex physics of the process needs to be taken into account. The advantage of PINN is that it minimizes a loss function during training, which takes into account the constraints of the system and th e laws of the domain. In this paper, we consider the solution of ordinary differential equations (ODEs) and PDEs using PINN, and then compare the efficiency and accuracy of this solution method compared to classical methods. The solution is implemented in the Julia programming language. We use NeuralPDE.jl, a package containing methods for solving equations in partial derivatives using physics-based neural networks. The classical method for solving PDEs is implemented through the DifferentialEquations.jl library. As a result, a comparative analysis of the considered solution methods for ODEs and PDEs has been performed, and an evaluation of their performance and accuracy has been obtained. In this paper we have demonstrated the basic capabilities of the NeuralPDE.jl package and its efficiency in comparison with numerical methods.