Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis
Stefano Berrone, Claudio Canuto, Moreno Pintore
Abstract
Abstract We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a posteriori error estimator, made of a residual-type term, a loss-function term, and data oscillation terms. We prove that the estimator is both reliable and efficient in controlling the energy norm of the error between the exact and VPINN solutions. Numerical results are in excellent agreement with the theoretical predictions.
Topics & Concepts
DiscretizationEstimatorApplied mathematicsA priori and a posterioriMathematicsNorm (philosophy)PiecewiseTriangulationArtificial neural networkPiecewise linear functionResidualMathematical optimizationCalculus (dental)Mathematical analysisComputer scienceAlgorithmGeometryStatisticsMachine learningDentistryLawPhilosophyPolitical scienceEpistemologyMedicineModel Reduction and Neural NetworksMagnetic Properties and ApplicationsNumerical methods in engineering