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A decoupled physics-informed neural network for recovering a space-dependent force function in the wave equation from integral overdetermination data

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Abstract

In this paper, we study the inverse problem of approximating a space-dependent wave source in a one-dimensional wave equation. The problem is converted to a nonclassical second-order hyperbolic equation and is later reduced to the solution of a linear system of algebraic equations by employing the Ritz collocation method. The obtained Ritz approximation is used to train a physics-informed neural network (PINN). However, unlike the conventional training in PINN, the loss function of the proposed method is not additive and the loss terms are decoupled in the sense that the neural network is re-trained separately according to each loss term. Hence, we name the proposed method decoupled PINN or D-PINN for short. Since the function of the initial condition is nonlinear and continuously differentiable, it is used, up to a scalar, as the activation function as opposed to using an off-the-shelf activation function such as hyperbolic tangent. Experiments indicate that D-PINN is capable of approximating the solution more accurately than Ritz and PINN.

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The data is generated by a numerical method and it is available.

Notes

  1. In this work we represent the transpose operator by notation tr.

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Acknowledgements

The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have led to improvements of the paper.

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Authors and Affiliations

  1. Global Artificial Intelligence Accelerator, Ericsson, Montreal, QC, H4S 0B6, Canada

    Aydin Sarraf

  2. School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

    Fatemeh Baharifard

  3. Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

    Kamal Rashedi

Authors
  1. Aydin Sarraf
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  2. Fatemeh Baharifard
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  3. Kamal Rashedi
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Contributions

AS designed the numerical algorithm, implemented the algorithm, and conducted numerical experiments for the neural network approximation. FB worked on writing the introduction, and contributed to the main methodology (the neural network aspect). KR worked on the Ritz approximation of the inverse wave problems, and contributed to the main methodology (the Ritz aspect) and conducted numerical experiments for the Ritz approximation. All authors reviewed the manuscript.

Corresponding author

Correspondence to Aydin Sarraf.

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Communicated by Vinicius Albani.

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Sarraf, A., Baharifard, F. & Rashedi, K. A decoupled physics-informed neural network for recovering a space-dependent force function in the wave equation from integral overdetermination data. Comp. Appl. Math. 42, 178 (2023). https://doi.org/10.1007/s40314-023-02323-9

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  • DOI: https://doi.org/10.1007/s40314-023-02323-9

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