DeepXDE 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; PyTorch; FPINNs; DiffSharp; TensorFlow; DeepONet; PDE-Net; NSFnets; PPINN; XPINNs; L-BFGS; PINNsNTK; PhyGeoNet; SciANN; SimNet; FEniCS; NeuroDiffEq; VarNet; PyDEns Cited in: 80 Publications all top 5 Cited by 238 Authors 13 Karniadakis, George Em 7 Meng, Xuhui 6 Lu, Lu 3 Cai, Shengze 3 Mao, Zhiping 3 Wang, Jianxun 3 Yan, Zhenya 2 Brunton, Steven L. 2 Gao, Han 2 Haghighat, Ehsan 2 Hwang, Hyung Ju 2 Jiao, Yuling 2 Kutz, J. Nathan 2 Lai, Yanming 2 Li, Dingwei 2 Lu, Xiliang 2 Psaros, Apostolos F. 2 Rabczuk, Timon 2 Raslan, Mones 2 Schneider, Reinhold 2 Siettos, Constantinos I. 2 Wang, Zhi Cheng 2 Yang, Jerry Zhijian 2 Zaki, Tamer A. 2 Zhang, Zhongqiang 2 Zhu, Xueyu 1 Alajlan, Naif 1 Amini Niaki, Sina 1 Amini, Danial 1 Anitescu, Cosmin 1 Aristotelous, Andreas C. 1 Ayensa-Jiménez, Jacobo 1 Baharifard, Fatemeh 1 Bai, Genming 1 Barati Farimani, Amir 1 Bazant, Martin Z. 1 Bramburger, Jason J. 1 Breth, Leoni 1 Cai, Weihua 1 Calabró, Francesco 1 Campbell, Trevor 1 Charalampidis, Efstathios G. 1 Chen, XiaoQian 1 Chen, Zhen 1 Chiu, Pao-Hsiung 1 Cho, Sung Woong 1 Choi, Youngsoo 1 Churchill, Victor 1 Cuomo, Salvatore 1 Dai, Chaoqing 1 Dao, My Ha 1 Darbon, Jerome 1 Deng, Xiangyun 1 Doblaré, Manuel 1 Dong, Suchuan 1 Doweidar, Mohamed Hamdy 1 Du, Qiuwan 1 Duan, Chenguang 1 Duchkov, Anton A. 1 Durlofsky, Louis J. 1 Eigel, Martin 1 Exl, Lukas 1 Fabiani, Gianluca 1 Fang, Yin 1 Fischbacher, Johann 1 Furati, Khaled M. 1 Gao, Jinghuai 1 Geist, Moritz 1 Ghodsi, Ali 1 Giampaolo, Fabio 1 Gong, Shibo 1 Goswami, Somdatta 1 Grubas, Serafim 1 Gu, Renshu 1 Gühring, Ingo 1 Guo, Hailong 1 Guo, Ling 1 Gusenbauer, Markus 1 Hajimohammadi, Zeinab 1 Hao, Shuo 1 He, Cuiyu 1 He, Juncai 1 Hernandez-Lopez, Francisco 1 Hill, Chris 1 Hovorka, Markus 1 Huang, Juntao 1 Itzá-Balam, Reymundo 1 Jang, Jin Woo 1 Jin, Xiaowei 1 Johnson, Steven G. 1 Juanes, Ruben 1 Kaheman, Kadierdan 1 Kato, Akira 1 Kawaguchi, Kenji 1 Kevrekidis, Ioannis George 1 Kevrekidis, Panayotis G. 1 Khademi, Wesley 1 Khaliq, Abdul Q. M. 1 Kharazmi, Ehsan 1 Kim, Youngkyu ...and 138 more Authors all top 5 Cited in 27 Serials 22 Computer Methods in Applied Mechanics and Engineering 22 Journal of Computational Physics 5 Communications in Computational Physics 4 Physica D 3 Journal of Scientific Computing 2 Physics Letters. A 2 Chaos, Solitons and Fractals 1 Computers & Mathematics with Applications 1 Mathematics and Computers in Simulation 1 Computer Aided Geometric Design 1 Journal of Computational Mathematics 1 Neural Networks 1 Applied Mathematical Modelling 1 International Journal of Computer Mathematics 1 SIAM Review 1 SIAM Journal on Scientific Computing 1 Fractals 1 ETNA. 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