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

By Jeremy Vincent Yu, Lu Lu, Xuhui Meng, G. Em Karniadakis. Computer Methods in Applied Mechanics and Engineering, 2021

771 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Electronic, Optical and Magnetic Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/669cc3240fff405b6ca87bd722b3a9dfe6adcb5c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. G. Em Karniadakis
2. Zhiping Mao
3. Lu Lu
4. Zhongqiang Zhang
5. Shengze Cai
6. Xuhui Meng
7. Pengzhan Jin
8. Shuai Meng

In Electronic, Optical and Magnetic Materials

1. G. Em Karniadakis
2. Lu Lu
3. Pengzhan Jin
4. Shuai Meng
5. Zhiping Mao
6. Xuhui Meng
7. Zhongqiang Zhang
8. Chen-Chun Wu

In Statistical and Nonlinear Physics

1. G. Em Karniadakis
2. Lu Lu
3. Zhiping Mao
4. Pengzhan Jin
5. Xuhui Meng
6. Shuai Meng
7. Shengze Cai
8. Zhongqiang Zhang

Citations in the corpus

• MIONet: Learning multiple-input operators via tensor product (2022, methodology): “As an active area of research in SciML, different methods have been developed to solve ordinary and partial differential equations (ODEs and PDEs) by parameterizing the solutions via neural networks (NNs), such as physics-informed NNs (PINNs) [48, 30, 35, 37, 47], deep Ritz method [43], and deep Gal”
• A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, background): “Physics-informed neural networks (PINNs) were introduced first in 2017 [7, 8] for forward, inverse and hybrid problems, and since then there have also been rapid developments in this area [9, 10, 11, 12, 13, 14, 15, 16, 17], although the achievements so far in scientific and engineering applications”
• A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks (2022, background): “For certain PDE problems, it is possible to enforce boundary conditions directly by constructing a special network architecture [27, 28, 8, 12], which eliminates the loss term of boundary conditions.”