  Physics-informed neural networks (PINNs) are one popular approach to
incorporate a priori knowledge about physical systems into the learning
framework. PINNs are known to be robust for smaller training sets, derive
better generalization problems, and are faster to train. In this paper, we show
that using PINNs in comparison with purely data-driven neural networks is not
only favorable for training performance but allows us to extract significant
information on the quality of the approximated solution. Assuming that the
underlying differential equation for the PINN training is an ordinary
differential equation, we derive a rigorous upper limit on the PINN prediction
error. This bound is applicable even for input data not included in the
training phase and without any prior knowledge about the true solution.
Therefore, our a posteriori error estimation is an essential step to certify
the PINN. We apply our error estimator exemplarily to two academic toy
problems, whereof one falls in the category of model-predictive control and
thereby shows the practical use of the derived results.
