  Physics informed neural networks (PINNs) have recently been widely used for
robust and accurate approximation of PDEs. We provide rigorous upper bounds on
the generalization error of PINNs approximating solutions of the forward
problem for PDEs. An abstract formalism is introduced and stability properties
of the underlying PDE are leveraged to derive an estimate for the
generalization error in terms of the training error and number of training
samples. This abstract framework is illustrated with several examples of
nonlinear PDEs. Numerical experiments, validating the proposed theory, are also
presented.
