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We consider the discretization of elliptic boundary-value problems byvariational physics-informed neural networks (VPINNs), in which test functionsare continuous, piecewise linear functions on a triangulation of the domain. Wedefine an a posteriori error estimator, made of a residual-type term, aloss-function term, and data oscillation terms. We prove that the estimator isboth reliable and efficient in controlling the energy norm of the error betweenthe exact and VPINN solutions. Numerical results are in excellent agreementwith the theoretical predictions.

Solving PDEs by Variational Physics-Informed Neural Networks: an a posteriori error analysis