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

Nature Machine Intelligence volume 3, pages 218–229 (2021)Cite this article

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Abstract

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.

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Fig. 1: Illustrations of the problem set-up and new architectures of DeepONets that lead to good generalization.
Fig. 2: Learning explicit operators using different V spaces and different network architectures.
Fig. 3: Fast learning of implicit operators in a nonlinear pendulum (k = 1 and T = 3).
Fig. 4: Fast learning of implicit operators in a diffusion-reaction system.
Fig. 5: DeepONet prediction for a stochastic ODE.
Fig. 6: DeepONet prediction for a stochastic elliptic equation.

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Data availability

All the datasets in the study were generated directly from the code.

Code availability

The code used in the study is publicly available from the GitHub repository https://github.com/lululxvi/deeponet55.

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Acknowledgements

This work was supported by the DOE PhILMs project (no. DE-SC0019453) and DARPA-CompMods grant no. HR00112090062.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Lu Lu

  2. Division of Applied Mathematics, Brown University, Providence, RI, USA

    Pengzhan Jin, Guofei Pang & George Em Karniadakis

  3. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Pengzhan Jin

  4. Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, USA

    Zhongqiang Zhang

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Contributions

L.L. and G.E.K. designed the study based on G.E.K.’s original idea. L.L. developed DeepONet architectures. L.L., P.J. and Z.Z. developed the theory. L.L. performed the experiments for the integral, nonlinear ODE, gravity pendulum and stochastic ODE/PDE operators. L.L. and P.J. performed the experiments for the Legendre transform, diffusion-reaction, advection and advection-diffusion PDEs. G.P. performed the experiments for fractional operators. L.L., P.J., G.P., Z.Z. and G.E.K. wrote the manuscript. G.E.K. supervised the project.

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Correspondence to George Em Karniadakis.

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The authors declare no competing interests.

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Peer review information Nature Machine Intelligence thanks Irana Higgins, Jian-Xun Wang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Lu, L., Jin, P., Pang, G. et al. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 3, 218–229 (2021). https://doi.org/10.1038/s42256-021-00302-5

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