DeepONet swMATH ID: 42093 Software Authors: Lu Lu, Pengzhan Jin, George Em Karniadakis Description: DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors xi,i=1,…,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size. Homepage: https://arxiv.org/abs/1910.03193 Source Code: https://github.com/lululxvi/deeponet Related Software: Adam; PDE-Net; DeepXDE; PyTorch; DGM; FPINNs; PINNsNTK; TensorFlow; ImageNet; XPINNs; NSFnets; GitHub; DiffSharp; AlexNet; Chebfun; torchdiffeq; MgNet; L-BFGS; Wasserstein GAN; JAX Cited in: 59 Publications all top 5 Cited by 170 Authors 9 Karniadakis, George Em 5 Perdikaris, Paris G. 4 Lu, Lu 4 Mao, Zhiping 4 Wang, Sifan 3 Meng, Xuhui 3 Zaki, Tamer A. 2 Cai, Shengze 2 Cyr, Eric C. 2 Goswami, Somdatta 2 Henkes, Alexander 2 Mahnken, Rolf 2 Mishra, Siddhartha 2 Molinaro, Roberto 2 Patel, Ravi G. 2 Schwab, Christoph 2 Stuart, Andrew M. 2 Tang, Yifa 2 Trask, Nathaniel A. 2 Wang, Hanwen 2 Wang, Jianxun 2 Wood, Mitchell A. 2 Yu, Yue 2 Zhu, Aiqing 1 Ainsworth, Mark 1 Bai, Genming 1 Bao, Gang 1 Bölcskei, Helmut 1 Bordas, Stéphane Pierre Alain 1 Bostanabad, Ramin 1 Breth, Leoni 1 Brunton, Steven L. 1 Budišić, Marko 1 Burkovska, Olena 1 Caylak, Ismail 1 Chandramowlishwaran, Aparna 1 Chen, Zhen 1 Choi, Youngsoo 1 Churchill, Victor 1 Clark di Leoni, Patricio 1 Cui, Tao 1 Cuomo, Salvatore 1 del Águila Ferrandis, José 1 D’Elia, Marta 1 Deparis, Simone 1 Deshpande, Saurabh A. 1 Dingreville, Rémi 1 Dong, Suchuan 1 Exl, Lukas 1 Fischbacher, Johann 1 Funke, Simon W. 1 Gao, Han 1 Gao, Yihang 1 Ghosh, Susanta 1 Giampaolo, Fabio 1 Gifford, Wesley M. 1 Giovanis, Dimitrios G. 1 Gong, Shibo 1 Gül, Recep 1 Gupta, Rachit 1 Gusenbauer, Markus 1 Han, Jiequn 1 He, Juncai 1 Herrmann, Lukas 1 Hong, Liu 1 Hosseini, Bamdad 1 Hovorka, Markus 1 Hu, Jiawei 1 Hu, Pipi 1 Hutter, Clemens 1 Jaiman, Rajeev Kumar 1 Ji, Tingwei 1 Jin, Pengzhan 1 Kaiser, Eurika 1 Kato, Akira 1 Kim, Youngkyu 1 Kinoshita, Akihito 1 Kissas, Georgios 1 Koley, Ujjwal 1 Kontolati, Katiana 1 Kornell, Alexander 1 Kovachki, Nikola B. 1 Kovács, Alexander 1 Kuchta, Miroslav 1 Kutz, J. Nathan 1 Lee, Chung-Hao 1 Lee, Myoungkyu 1 Lengiewicz, Jakub 1 Lessig, Christian 1 Li, Zan 1 Li, Zhen 1 Li, Zongwei 1 Lin, Chensen 1 Liu, Jian 1 Liu, Yang 1 Loukrezis, Dimitrios 1 Lye, Kjetil Olsen 1 Manickam, Indu 1 Margenberg, Nils 1 Marxen, Olaf ...and 70 more Authors all top 5 Cited in 20 Serials 17 Journal of Computational Physics 15 Computer Methods in Applied Mechanics and Engineering 4 SIAM Journal on Scientific Computing 3 Computers & Mathematics with Applications 3 Journal of Scientific Computing 2 Journal of Fluid Mechanics 2 SIAM Review 1 Computers and Fluids 1 Inverse Problems 1 Journal of Computational and Applied Mathematics 1 Journal of Computational Mathematics 1 Physica D 1 Neural Networks 1 European Journal of Applied Mathematics 1 SIAM Journal on Applied Mathematics 1 Applied and Computational Harmonic Analysis 1 ETNA. Electronic Transactions on Numerical Analysis 1 Communications in Nonlinear Science and Numerical Simulation 1 Research in the Mathematical Sciences 1 SMAI Journal of Computational Mathematics all top 5 Cited in 21 Fields 42 Computer science (68-XX) 35 Numerical analysis (65-XX) 16 Partial differential equations (35-XX) 11 Fluid mechanics (76-XX) 7 Statistics (62-XX) 6 Biology and other natural sciences (92-XX) 5 Mechanics of deformable solids (74-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Approximations and expansions (41-XX) 3 Geophysics (86-XX) 2 Operator theory (47-XX) 2 Statistical mechanics, structure of matter (82-XX) 2 Operations research, mathematical programming (90-XX) 2 Systems theory; control (93-XX) 1 Real functions (26-XX) 1 Ordinary differential equations (34-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Probability theory and stochastic processes (60-XX) 1 Mechanics of particles and systems (70-XX) 1 Quantum theory (81-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) Citations by Year