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; DGM; DeepXDE; PyTorch; PDE-Net; TensorFlow; DiffSharp; FPINNs; XPINNs; hp-VPINNs; ImageNet; PINNsNTK; PhyGeoNet; AlexNet; NSFnets; FEniCS; torchdiffeq; GitHub; redbKIT; L-BFGS
Cited in: 255 Documents
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Cited by 639 Authors

23 Karniadakis, George Em
9 Lu, Lu
8 Perdikaris, Paris G.
6 Mao, Zhiping
6 Meng, Xuhui
6 Mishra, Siddhartha
6 Yu, Yue
5 Goswami, Somdatta
5 Manzoni, Andrea
5 Wang, Sifan
4 Franco, Nicola Rares
4 Ghattas, Omar N.
4 Ma, Zheng
4 O’Leary-Roseberry, Thomas
4 Stuart, Andrew M.
4 Yan, Zhenya
4 Zaki, Tamer A.
3 Chakraborty, Souvik
3 Churchill, Victor
3 Cui, Tiangang
3 Cyr, Eric C.
3 Darbon, Jerome
3 De Ryck, Tim
3 D’Elia, Marta
3 Fresca, Stefania
3 Han, Jiequn
3 Hosseini, Bamdad
3 Hu, Pipi
3 Jin, Pengzhan
3 Jin, Shi
3 Kovachki, Nikola B.
3 Kutz, J. Nathan
3 Li, Qianxiao
3 Lin, Guang
3 Margenberg, Nils
3 Molinaro, Roberto
3 Rozza, Gianluigi
3 Schwab, Christoph
3 Shin, Yeonjong
3 Stinis, Panos
3 Tang, Yifa
3 Trask, Nathaniel A.
3 Wang, Jianxun
3 Xiu, Dongbin
3 You, Huaiqian
3 Zhang, Zhongqiang
3 Zhong, Ming
3 Zhu, Aiqing
3 Zunino, Paolo
2 Aldirany, Ziad
2 Brunton, Steven L.
2 Cai, Shengze
2 Chen, Fukai
2 Chen, Junqing
2 Chen, Peng
2 Chen, Yong
2 Chen, Zhen
2 Clark di Leoni, Patricio
2 Cottereau, Régis
2 Demo, Nicola
2 Dietrich, Felix
2 Dolgov, Sergey V.
2 Dong, Huanhe
2 E, Weinan
2 Gao, Tian
2 Haghighat, Ehsan
2 Hao, Wenrui
2 Henkes, Alexander
2 Howard, Amanda A.
2 Huang, Haoxiang
2 Jha, Prashant Kumar
2 Ju, Lili
2 Kevrekidis, Ioannis George
2 Kontolati, Katiana
2 Kuchta, Miroslav
2 Laforest, Marc
2 Lanthaler, Samuel
2 Lee, Chung-Hao
2 Lessig, Christian
2 Li, Zhen
2 Lin, Guochang
2 Liu, Yang
2 Lye, Kjetil Olsen
2 Mahnken, Rolf
2 Meneveau, Charles
2 Meng, Tingwei
2 N., Navaneeth
2 Nelsen, Nicholas H.
2 Ning, Jianfeng
2 Oberai, Assad A.
2 Owhadi, Houman
2 Patel, Ravi G.
2 Perego, Mauro
2 Prudhomme, Serge
2 Psaros, Apostolos F.
2 Rao, Chengping
2 Ray, Deep
2 Ren, Pu
2 Schäfer, Florian
2 Schiavazzi, Daniele E.
...and 539 more Authors
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Cited in 54 Serials

69 Journal of Computational Physics
56 Computer Methods in Applied Mechanics and Engineering
15 Journal of Scientific Computing
12 SIAM Journal on Scientific Computing
8 Physica D
8 Neural Networks
6 Computers & Mathematics with Applications
6 Computational Mechanics
6 Communications in Computational Physics
4 Journal of Computational and Applied Mathematics
4 Advances in Computational Mathematics
3 Inverse Problems
3 SIAM Review
2 Journal of Fluid Mechanics
2 Journal of Mathematical Biology
2 SIAM Journal on Numerical Analysis
2 Journal of Computational Mathematics
2 Japan Journal of Industrial and Applied Mathematics
2 Applied and Computational Harmonic Analysis
2 Communications in Nonlinear Science and Numerical Simulation
2 Journal of Systems Science and Complexity
2 Journal of Machine Learning Research (JMLR)
2 SIAM Journal on Applied Dynamical Systems
2 AMM. Applied Mathematics and Mechanics. (English Edition)
1 Computers and Fluids
1 Journal of Mathematical Analysis and Applications
1 Mathematics of Computation
1 Bulletin of Mathematical Biology
1 Chaos, Solitons and Fractals
1 International Journal for Numerical Methods in Engineering
1 Systems & Control Letters
1 MCSS. Mathematics of Control, Signals, and Systems
1 European Journal of Applied Mathematics
1 SIAM Journal on Applied Mathematics
1 SIAM Journal on Mathematical Analysis
1 ETNA. Electronic Transactions on Numerical Analysis
1 Annals of Mathematics and Artificial Intelligence
1 Engineering Analysis with Boundary Elements
1 Doklady Mathematics
1 International Journal of Computational Fluid Dynamics
1 Mathematical Finance
1 Nonlinear Dynamics
1 European Journal of Mechanics. A. Solids
1 Stochastics and Dynamics
1 International Journal of Computational Methods
1 Acta Numerica
1 GAMM-Mitteilungen
1 Communications in Theoretical Physics
1 Communications in Mathematics and Statistics
1 SIAM/ASA Journal on Uncertainty Quantification
1 Research in the Mathematical Sciences
1 Journal of Nonlinear and Variational Analysis
1 SMAI Journal of Computational Mathematics
1 Communications of the American Mathematical Society

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