DeepXDE

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 38 articles )

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  1. 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)
  2. 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)
  3. Li, Liangliang; Li, Yunzhu; Du, Qiuwan; Liu, Tianyuan; Xie, Yonghui: ReF-nets: physics-informed neural network for Reynolds equation of gas bearing (2022)
  4. Mo, Yifan; Ling, Liming; Zeng, Delu: Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm (2022)
  5. Ren, Pu; Rao, Chengping; Liu, Yang; Wang, Jian-Xun; Sun, Hao: PhyCRNet: physics-informed convolutional-recurrent network for solving spatiotemporal PDEs (2022)
  6. Sukumar, N.; Srivastava, Ankit: Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2022)
  7. Amini Niaki, Sina; Haghighat, Ehsan; Campbell, Trevor; Poursartip, Anoush; Vaziri, Reza: Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture (2021)
  8. Ayensa-Jiménez, Jacobo; Doweidar, Mohamed H.; Sanz-Herrera, Jose A.; Doblaré, Manuel: Prediction and identification of physical systems by means of physically-guided neural networks with meaningful internal layers (2021)
  9. Bramburger, Jason J.; Brunton, Steven L.; Nathan Kutz, J.: Deep learning of conjugate mappings (2021)
  10. Calabrò, Francesco; Fabiani, Gianluca; Siettos, Constantinos: Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients (2021)
  11. Cho, Sung Woong; Hwang, Hyung Ju; Son, Hwijae: Traveling wave solutions of partial differential equations via neural networks (2021)
  12. Geist, Moritz; Petersen, Philipp; Raslan, Mones; Schneider, Reinhold; Kutyniok, Gitta: Numerical solution of the parametric diffusion equation by deep neural networks (2021)
  13. Gühring, Ingo; Raslan, Mones: Approximation rates for neural networks with encodable weights in smoothness spaces (2021)
  14. Itzá Balam, Reymundo; Hernandez-Lopez, Francisco; Trejo-Sánchez, Joel; Uh Zapata, Miguel: An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains (2021)
  15. Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M.: \textithp-VPINNs: variational physics-informed neural networks with domain decomposition (2021)
  16. Lee, Jae Yong; Jang, Jin Woo; Hwang, Hyung Ju: The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the deep neural network approach (2021)
  17. Li, Wei; Bazant, Martin Z.; Zhu, Juner: A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches (2021)
  18. Long, Jie; Khaliq, A. Q. M.; Furati, K. M.: Identification and prediction of time-varying parameters of COVID-19 model: a data-driven deep learning approach (2021)
  19. Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em: DeepXDE: a deep learning library for solving differential equations (2021)
  20. Lu, Lu; Pestourie, Raphaël; Yao, Wenjie; Wang, Zhicheng; Verdugo, Francesc; Johnson, Steven G.: Physics-informed neural networks with hard constraints for inverse design (2021)

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