Deferred correction neural network techniques for solving ordinary differential equations

Abstract

Neural network technology is widely used to solve problems across science and engineering fields. In this paper, a neural network technique is applied to solve ordinary differential equations (ODEs), instead of using conventional time marching techniques with discretization for ODEs. A numerical solution to ODEs is defined with polynomial basis, and each coefficient of the expansion is calculated through an unsupervised neural network structure in the entire time domain. To make the calculated solution more accurate, we consider the deferred correction technique for calculating a numerical residual with a physics-informed neural network. The empirical results show that the proposed scheme with polynomial basis can influence calculation accuracy depending on the order of basis polynomials, providing the possibility of improving accuracy and efficiency by increasing the order of basis. Several numerical tests confirm that the proposed deferred correction network (DCNet) model has an accuracy approximately 100 and 10 times higher than that of the learning polynomial neural network (LPNet) and the standard existing method, physics-informed neural network (PINN), respectively. Additionally, the proposed schemes generate stable results even for stiff problems and preserve the conservation property for Hamiltonian systems.