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Control of Partial Differential Equations via Physics-Informed Neural Networks

Author

Listed:
  • Carlos J. García-Cervera

    (University of California)

  • Mathieu Kessler

    (Technical University of Cartagena)

  • Francisco Periago

    (Technical University of Cartagena)

Abstract

This paper addresses the numerical resolution of controllability problems for partial differential equations (PDEs) by using physics-informed neural networks. Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds, that apply to any exact controllable linear system of PDEs and in any dimension, provide a rigorous justification for the use of neural networks in this field. Preliminary numerical simulation results for three different types of PDEs are carried out to illustrate the performance of the proposed methodology.

Suggested Citation

  • Carlos J. García-Cervera & Mathieu Kessler & Francisco Periago, 2023. "Control of Partial Differential Equations via Physics-Informed Neural Networks," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 391-414, February.
    • Handle: RePEc:spr:joptap:v:196:y:2023:i:2:d:10.1007_s10957-022-02100-4
    DOI: 10.1007/s10957-022-02100-4
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    References listed on IDEAS

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    1. Francisco J. Marín & Jesús Martínez-Frutos & Francisco Periago, 2017. "Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 428-454, August.
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