Using Neural Networks with Numerical Methods to Solve Complex Differential Equations

Abstract

This research investigates the use of neural networks in combination with the numerical methods and their application to the solution of the complex differential equations. Many of the scientific and engineering phenomena that are commonly modeled with differential equations can't be solved analytically because of complex boundary conditions, high dimension, stiffness, or nonlinearity. Thus, this study aims to fairly compare the numerical methods with the physics-informed neural network and hybridized neural–numerical methods, in terms of their accuracy, convergence, stability and computational efficiency. The methodology used is analytical–computational using selected test problems with known exact solutions namely, an ordinary differential equation (ODE), a boundary value problem (BVP) and a partial differential equation (PDE). It shows that neural networks are efficient at giving flexible approximation in solving the problem while the classical numerical methods are efficient in solving structured problems. The hybrid neural-numerical method gives the best overall performance in which both the stable numerical method and learning ability of neural networks are combined. In this study the conclusion is drawn that a neural network should not be used to replace traditional methods, but should be implemented alongside, to complement.