Multilevel physics informed neural networks (MPINNs)

Elisa Riccietti, Valentin Mercier, Serge Gratton, Pierre Boudier

Abstract: In this paper we introduce multilevel physics informed neural networks (MPINNs). Inspired by classical multigrid methods for the solution of linear systems arising from the discretization of PDEs, our MPINNs are based on the classical correction scheme, which represents the solution as the sum of a fine and a coarse term that are optimized in an alternate way. We show that the proposed approach allows to reproduce in the neural network training the classical acceleration effect observed for classical multigrid methods, thus providing a PINN that shows improved performance compared to the state-of-the-art. Thanks to the support of the coarse model, MPINNs provide indeed a faster and improved decrease of the approximation error in the case both of elliptic and nonlinear equations.