Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

TL;DR

This paper introduces gradient-enhanced physics-informed neural networks (gPINNs), which incorporate gradient information of PDE residuals to improve accuracy and efficiency in solving forward and inverse PDE problems, especially with steep gradients.

Contribution

The paper proposes gPINNs that embed PDE residual gradients into the loss function, enhancing accuracy and training efficiency over traditional PINNs.

Findings

gPINNs outperform PINNs with fewer training points

gPINNs achieve higher accuracy in forward and inverse PDE problems

Combining gPINNs with adaptive refinement further improves results

Abstract

Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual into the loss function of the neural network, and have been successfully employed to solve diverse forward and inverse PDE problems. However, one disadvantage of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose a new method, gradient-enhanced physics-informed neural networks (gPINNs), for improving the accuracy and training efficiency of PINNs. gPINNs leverage gradient information of the PDE residual and embed the gradient into the loss function. We tested gPINNs extensively and demonstrated the effectiveness of gPINNs in both forward and inverse PDE problems. Our numerical results show that gPINN performs better than PINN with fewer training points. Furthermore, we combined gPINN with the method of residual-based adaptive refinement (RAR), a method for improving the distribution of training points adaptively during training, to further improve the performance of gPINN, especially in PDEs with solutions that have steep gradients.