Error estimation for physics-informed neural networks with implicit
  Runge-Kutta methods
Jochen Stiasny Spyros Chatzivasileiadis
The ability to accurately approximate trajectories of dynamical systems
enables their analysis, prediction, and control. Neural network (NN)-based
approximations have attracted significant interest due to fast evaluation with
good accuracy over long integration time steps. In contrast to established
numerical approximation schemes such as Runge-Kutta methods, the estimation of
the error of the NN-based approximations proves to be difficult. In this work,
we propose to use the NN's predictions in a high-order implicit Runge-Kutta
(IRK) method. The residuals in the implicit system of equations can be related
to the NN's prediction error, hence, we can provide an error estimate at
several points along a trajectory. We find that this error estimate highly
correlates with the NN's prediction error and that increasing the order of the
IRK method improves this estimate. We demonstrate this estimation methodology
for Physics-Informed Neural Network (PINNs) on the logistic equation as an
illustrative example and then apply it to a four-state electric generator model
that is regularly used in power system modelling.