Control of partial differential equations via physics-informed neural networks
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This research was supported by Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia (Spain)) under contract 20911/PI/18 and grant number 21503/EE/21 (mobility program Jiménez de la Espada). F. Periago acknowledges the hospitality of the Mathematics Department at University of California, Santa Barbara, where part of this work was carried out. The authors also thank professor Lu Lu for very fruitful comments on the use of DeepXDE.Realizado en/con
Universidad Politécnica de CartagenaFecha de publicación
2022-09-17Editorial
SPRINGERCita bibliográfica
García-Cervera, C.J., Kessler, M. & Periago, F. Control of Partial Differential Equations via Physics-Informed Neural Networks. J Optim Theory Appl 196, 391–414 (2023). https://doi.org/10.1007/s10957-022-02100-4Revisión por pares
SIPalabras clave
Controllability of partial differential equationsPhysics-informed neural networks
Error estimates
Resumen
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.
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