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Prediction error quantification in machine learning has been left out of mostmethodological investigations of neural networks, for both purely data-drivenand physics-informed approaches. Beyond statistical investigations and genericresults on the approximation capabilities of neural networks, we present arigorous upper bound on the prediction error of physics-informed neuralnetworks. This bound can be calculated without the knowledge of the truesolution and only with a priori available information about the characteristicsof the underlying dynamical system governed by a partial differential equation.We apply this a posteriori error bound exemplarily to four problems: thetransport equation, the heat equation, the Navier-Stokes equation and theKlein-Gordon equation.

Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs