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Higher-order error estimates for physics-informed neural networks approximating the primitive equations

Author

Listed:
  • Ruimeng Hu

    (University of California
    University of California)

  • Quyuan Lin

    (University of California
    Clemson University)

  • Alan Raydan

    (University of California)

  • Sui Tang

    (University of California)

Abstract

Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates are a priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the $$H^s$$ H s norm during the training.

Suggested Citation

  • Ruimeng Hu & Quyuan Lin & Alan Raydan & Sui Tang, 2023. "Higher-order error estimates for physics-informed neural networks approximating the primitive equations," Partial Differential Equations and Applications, Springer, vol. 4(4), pages 1-35, August.
    • Handle: RePEc:spr:pardea:v:4:y:2023:i:4:d:10.1007_s42985-023-00254-y
    DOI: 10.1007/s42985-023-00254-y
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    References listed on IDEAS

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    1. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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