Rigorous a Posteriori Error Bounds for PDE-Defined PINNs

Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks (NNs), for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of NNs, we present a rigorous upper bound on the prediction error of physics-informed NNs (PINNs). 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 (PDE). We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation (NSE), and the Klein-Gordon equation.