DeepXDE: A deep learning library for solving differential equations. Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.

References in zbMATH (referenced in 61 articles )

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  1. Chen, Zhen; Churchill, Victor; Wu, Kailiang; Xiu, Dongbin: Deep neural network modeling of unknown partial differential equations in nodal space (2022)
  2. Chiu, Pao-Hsiung; Wong, Jian Cheng; Ooi, Chinchun; Dao, My Ha; Ong, Yew-Soon: CAN-PINN: a fast physics-informed neural network based on coupled-automatic-numerical differentiation method (2022)
  3. Duan, Chenguang; Jiao, Yuling; Lai, Yanming; Li, Dingwei; Lu, Xiliang; Yang, Jerry Zhijian: Convergence rate analysis for deep Ritz method (2022)
  4. Gao, Han; Zahr, Matthew J.; Wang, Jian-Xun: Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems (2022)
  5. Guo, Hailong; Yang, Xu: Deep unfitted Nitsche method for elliptic interface problems (2022)
  6. Huang, Juntao; Zhou, Yizhou; Yong, Wen-An: Data-driven discovery of multiscale chemical reactions governed by the law of mass action (2022)
  7. Jiao, Yuling; Lai, Yanming; Li, Dingwei; Lu, Xiliang; Wang, Fengru; Wang, Yang; Yang, Jerry Zhijian: A rate of convergence of physics informed neural networks for the linear second order elliptic PDEs (2022)
  8. Kim, Youngkyu; Choi, Youngsoo; Widemann, David; Zohdi, Tarek: A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder (2022)
  9. Kovacs, Alexander; Exl, Lukas; Kornell, Alexander; Fischbacher, Johann; Hovorka, Markus; Gusenbauer, Markus; Breth, Leoni; Oezelt, Harald; Yano, Masao; Sakuma, Noritsugu; Kinoshita, Akihito; Shoji, Tetsuya; Kato, Akira; Schrefl, Thomas: Conditional physics informed neural networks (2022)
  10. Li, Liangliang; Li, Yunzhu; Du, Qiuwan; Liu, Tianyuan; Xie, Yonghui: ReF-nets: physics-informed neural network for Reynolds equation of gas bearing (2022)
  11. Lu, Lu; Meng, Xuhui; Cai, Shengze; Mao, Zhiping; Goswami, Somdatta; Zhang, Zhongqiang; Karniadakis, George Em: A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2022)
  12. Mo, Yifan; Ling, Liming; Zeng, Delu: Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm (2022)
  13. Psaros, Apostolos F.; Kawaguchi, Kenji; Karniadakis, George Em: Meta-learning PINN loss functions (2022)
  14. Qu, Jiagang; Cai, Weihua; Zhao, Yijun: Learning time-dependent PDEs with a linear and nonlinear separate convolutional neural network (2022)
  15. Ren, Pu; Rao, Chengping; Liu, Yang; Wang, Jian-Xun; Sun, Hao: PhyCRNet: physics-informed convolutional-recurrent network for solving spatiotemporal PDEs (2022)
  16. Rivera, Jon A.; Taylor, Jamie M.; Omella, Ángel J.; Pardo, David: On quadrature rules for solving partial differential equations using neural networks (2022)
  17. Sukumar, N.; Srivastava, Ankit: Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2022)
  18. Wang, Dandan; Xu, Jinlan; Gao, Fei; Wang, Charlie C. L.; Gu, Renshu; Lin, Fei; Rabczuk, Timon; Xu, Gang: IGA-reuse-NET: a deep-learning-based isogeometric analysis-reuse approach with topology-consistent parameterization (2022)
  19. Wang, Sifan; Yu, Xinling; Perdikaris, Paris: When and why PINNs fail to train: a neural tangent kernel perspective (2022)
  20. Yu, Jeremy; Lu, Lu; Meng, Xuhui; Karniadakis, George Em: Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems (2022)

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