Control of Partial Differential Equations via Physics-Informed Neural Networks

Carlos J. García-Cervera (), Mathieu Kessler () and Francisco Periago ()
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Carlos J. García-Cervera: University of California
Mathieu Kessler: Technical University of Cartagena
Francisco Periago: Technical University of Cartagena

Journal of Optimization Theory and Applications, 2023, vol. 196, issue 2, No 1, 414 pages

Abstract: 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.

Keywords: Controllability of partial differential equations; Physics-informed neural networks; Error estimates; 93B05; 93C20; 65N15 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-022-02100-4

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