We introduce a Robust version of the Variational Physics-Informed Neural
Networks (RVPINNs) to approximate the Partial Differential Equations (PDEs)
solution. We start from a weak Petrov-Galerkin formulation of the problem,
select a discrete test space, and define a quadratic loss functional as in
VPINNs. Whereas in VPINNs the loss depends upon the selected basis functions of
a given test space, herein we minimize a loss based on the residual in the
discrete dual norm, which is independent of the test space's choice of test
basis functions. We demonstrate that this loss is a reliable and efficient
estimator of the true error in the energy norm. The proposed loss function
requires computation of the Gram matrix inverse, similar to what occurs in
traditional residual minimization methods. To validate our theoretical
findings, we test the performance and robustness of our algorithm in several
advection-dominated-diffusion problems in one spatial dimension. We conclude
that RVPINNs is a robust method