  Physics informed neural networks (PINNs) have recently been very successfully
applied for efficiently approximating inverse problems for PDEs. We focus on a
particular class of inverse problems, the so-called data assimilation or unique
continuation problems, and prove rigorous estimates on the generalization error
of PINNs approximating them. An abstract framework is presented and conditional
stability estimates for the underlying inverse problem are employed to derive
the estimate on the PINN generalization error, providing rigorous justification
for the use of PINNs in this context. The abstract framework is illustrated
with examples of four prototypical linear PDEs. Numerical experiments,
validating the proposed theory, are also presented.
