Control of Partial Differential Equations via Physics-Informed Neural Networks
Carlos J. Garcı́a-Cervera, Mathieu Kessler, Francisco Periago
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.
Topics & Concepts
ObservabilityControllabilityPartial differential equationArtificial neural networkMathematicsTheory of computationDimension (graph theory)Applied mathematicsGeneralizationOptimal controlMathematical optimizationComputer scienceAlgorithmMathematical analysisArtificial intelligencePure mathematicsModel Reduction and Neural NetworksStability and Controllability of Differential EquationsControl Systems and Identification