swMATH ID: 32456
Software Authors: Lu Lu, Xuhui Meng, Zhiping Mao, George E. Karniadakis
Description: 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.
Homepage: https://arxiv.org/abs/1907.04502
Source Code:  https://github.com/lululxvi/deepxde
Related Software: Adam; DGM; FPINNs; DeepONet; PyTorch; DiffSharp; TensorFlow; XPINNs; PINNsNTK; hp-VPINNs; L-BFGS; PPINN; NSFnets; PhyGeoNet; SciANN; PDE-Net; AlexNet; ImageNet; SimNet; NeuroDiffEq
Cited in: 206 Documents
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Cited by 556 Authors

18 Karniadakis, George Em
10 Lu, Lu
9 Meng, Xuhui
9 Yan, Zhenya
5 Dong, Suchuan
5 Jiao, Yuling
3 Cai, Shengze
3 Chen, Yong
3 Churchill, Victor
3 Hwang, Hyung Ju
3 Lai, Yanming
3 Li, Wei
3 Lu, Xiliang
3 Mao, Zhiping
3 Margenberg, Nils
3 Perdikaris, Paris G.
3 Psaros, Apostolos F.
3 Wang, Jianxun
3 Wang, Li
3 Wang, Sifan
3 Xiu, Dongbin
3 Yang, Jerry Zhijian
3 Yang, Xu
3 Zhong, Ming
3 Zhou, Zijian
3 Zou, Zongren
2 Bazant, Martin Z.
2 Brunton, Steven L.
2 Bui-Thanh, Tan
2 Chen, Zhen
2 Diao, Yu
2 Duan, Jinqiao
2 Feng, Xinlong
2 Gao, Han
2 Gu, Tongxiang
2 Guo, Jiawei
2 Guo, Ling
2 Haghighat, Ehsan
2 Jentzen, Arnulf
2 Jin, Shi
2 Kutz, J. Nathan
2 Lessig, Christian
2 Li, Dingwei
2 Li, Jian
2 Li, Zhen
2 Liu, Yinghua
2 Lorin, Emmanuel
2 Ma, Zheng
2 Mishra, Siddhartha
2 Nguyen, Hai V.
2 Pardo, David
2 Petersen, Philipp Christian
2 Rabczuk, Timon
2 Raslan, Mones
2 Schneider, Reinhold
2 Siettos, Constantinos I.
2 Son, Hwijae
2 Sun, Jia
2 Taylor, Jamie M.
2 Treister, Eran
2 Wang, Fengru
2 Wang, Yang
2 Wang, Yiran
2 Wang, Yizheng
2 Wang, Zhi Cheng
2 Wu, Keke
2 Xu, Hui
2 Yan, Liang
2 Yang, Jianchuan
2 Yang, Jiang
2 Yao, Yanzhong
2 Zaki, Tamer A.
2 Zhang, Wen
2 Zhang, Zhiyong
2 Zhang, Zhongqiang
2 Zhou, Tao
2 Zhu, Juner
2 Zhu, Min
2 Zhu, Quanhui
2 Zhu, Xueyu
2 Zohdi, Tarek I.
1 Abueidda, Diab W.
1 Alajlan, Naif
1 Aldakheel, Fadi
1 Alfaro, Iciar
1 Amini Niaki, Sina
1 Amini, Danial
1 Anagnostopoulos, Sokratis J.
1 Anitescu, Cosmin
1 Aristotelous, Andreas C.
1 Ayensa-Jiménez, Jacobo
1 Azulay, Yael
1 Badías, Alberto
1 Baek, Jonghyuk
1 Baharifard, Fatemeh
1 Bai, Genming
1 Bao, Feng
1 Barati Farimani, Amir
1 Bause, Markus
1 Beck, Christian
...and 456 more Authors
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Cited in 55 Serials

46 Computer Methods in Applied Mechanics and Engineering
46 Journal of Computational Physics
12 Physica D
10 Journal of Scientific Computing
10 Communications in Computational Physics
8 SIAM Journal on Scientific Computing
6 Computers & Mathematics with Applications
3 Physics Letters. A
3 Chaos, Solitons and Fractals
3 Journal of Computational and Applied Mathematics
3 Computational Mechanics
3 Applied Mathematical Modelling
3 Advances in Computational Mathematics
3 Numerical Mathematics: Theory, Methods and Applications
2 International Journal for Numerical Methods in Engineering
2 International Journal of Computer Mathematics
2 SIAM Review
2 Mathematical Biosciences and Engineering
2 Communications in Theoretical Physics
1 Computers and Fluids
1 Computer Physics Communications
1 Inverse Problems
1 Physica A
1 Bulletin of Mathematical Biology
1 Journal of Optimization Theory and Applications
1 Mathematics and Computers in Simulation
1 Computer Aided Geometric Design
1 Journal of Computational Mathematics
1 Applied Mathematics Letters
1 Neural Networks
1 Fractals
1 ETNA. Electronic Transactions on Numerical Analysis
1 Mathematical Problems in Engineering
1 Doklady Mathematics
1 International Journal of Computational Fluid Dynamics
1 Mathematical Finance
1 Nonlinear Dynamics
1 Communications in Nonlinear Science and Numerical Simulation
1 Nonlinear Analysis. Modelling and Control
1 Quantitative Finance
1 Foundations of Computational Mathematics
1 Discrete and Continuous Dynamical Systems. Series B
1 SIAM Journal on Applied Dynamical Systems
1 Analysis and Applications (Singapore)
1 International Journal of Numerical Analysis and Modeling
1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis
1 Probability Surveys
1 East Asian Journal on Applied Mathematics
1 SIAM/ASA Journal on Uncertainty Quantification
1 Journal of Mathematical Modeling
1 AMM. Applied Mathematics and Mechanics. (English Edition)
1 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences
1 Journal of Nonlinear and Variational Analysis
1 Communications on Applied Mathematics and Computation
1 SN Partial Differential Equations and Applications

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