  Prediction error quantification in machine learning has been left out of most
methodological investigations of neural networks, for both purely data-driven
and physics-informed approaches. Beyond statistical investigations and generic
results on the approximation capabilities of neural networks, we present a
rigorous upper bound on the prediction error of physics-informed neural
networks. This bound can be calculated without the knowledge of the true
solution and only with a priori available information about the characteristics
of the underlying dynamical system governed by a partial differential equation.
We apply this a posteriori error bound exemplarily to four problems: the
transport equation, the heat equation, the Navier-Stokes equation and the
Klein-Gordon equation.
