Physics-informed neural networks with adaptive loss weighting algorithm for solving partial differential equations

Abstract

In recent years, physics-informed neural networks (PINNs) have garnered widespread attentions for the ability of solving nonlinear partial differential equations (PDE) using neural networks. The paper regards PINNs as multitask learning and proposes an adaptive loss weighting algorithm in physics-informed neural networks (APINNs). APINNs could balance the magnitudes of different loss functions during the training process to ensure a balanced contribution of parameters with different magnitudes to loss functions, thereby training solutions that satisfy initial boundary conditions and physical equations. Based on the original PINNs and APINNs, we respectively simulated the solitary wave solution of the Benjamin-Ono equation, the breather wave solution of the Sine-Gordon equation and the breather wave solution of the Mukherjee-Kundu equation. In the experiment of solving the solitary wave solution of the Benjamin-Ono equation, the minimum predict error of PINNs is about 60%, while the minimum predict error of APINNs is about 1%. As for solving breather wave solution, for the Sine-Gordon equation the minimum predict error of PINNs is around 10%, while the minimum predict error of APINNs is around 2%; and for the Mukherjee-Kundu equation, the minimum predict error of PINNs is about 30%, while the minimum predict error of APINNs is about 10%. The experimental results show that compared with PINNs, the predict solutions trained by APINNs have smaller errors.