  We analyze neural network solutions to partial differential equations
obtained with Physics Informed Neural Networks. In particular, we apply tools
of classical finite element error analysis to obtain conclusions about the
error of the Deep Ritz method applied to the Laplace and the Stokes equations.
Further, we develop an a posteriori error estimator for neural network
approximations of partial differential equations. The proposed approach is
based on the dual weighted residual estimator. It is destined to serve as a
stopping criterion that guarantees the accuracy of the solution independently
of the design of the neural network training. The result is equipped with
computational examples for Laplace and Stokes problems.
