Introducing the Picard Method for Approximating Solutions of Differential Equations with Neural Networks

Abstract

For decades, differential equations have been used to model various problems in the natural sciences and engineering. However, ordinary differential equations (ODEs) are usually not analytically solvable, so many numerical approaches have been developed to produce approximate solutions. More recently, it has been proposed that neural networks can learn solutions of ODEs and thus provide faster and more accurate numerical approximations. Here, we propose a novel approach of having neural networks learn solutions to ODEs via the Picard formulation. We show, through examples, that this approach produces approximations that are at least as reliable as earlier approaches.