Control of Partial Differential Equations via Physics-Informed Neural Networks

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.