Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is sugg...

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Veröffentlicht in:Nature machine intelligence Jg. 3; H. 3; S. 218 - 229
Hauptverfasser: Lu, Lu, Jin, Pengzhan, Pang, Guofei, Zhang, Zhongqiang, Karniadakis, George Em
Format: Journal Article
Sprache:Englisch
Veröffentlicht: London Nature Publishing Group UK 01.03.2021
Nature Publishing Group
The Author(s), Springer Nature
Schlagworte:
639/705/1041
639/705/1042
Accuracy
Approximation
Artificial neural networks
Banach spaces
Complex systems
Computer Science
Continuity (mathematics)
Deep learning
Differential equations
Dynamical systems
Engineering
Errors
Function space
Machine learning
Mathematical analysis
Neural networks
Operators (mathematics)
Optimization
Ordinary differential equations
Partial differential equations
Robotics
Theorems
ISSN:2522-5839, 2522-5839
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Zusammenfassung:It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.
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SC0019453
USDOE Office of Science (SC)
ISSN:2522-5839
2522-5839
DOI:10.1038/s42256-021-00302-5