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Physics informed neural networks (PINNs) have recently been very successfullyapplied for efficiently approximating inverse problems for PDEs. We focus on aparticular class of inverse problems, the so-called data assimilation or uniquecontinuation problems, and prove rigorous estimates on the generalization errorof PINNs approximating them. An abstract framework is presented and conditionalstability estimates for the underlying inverse problem are employed to derivethe estimate on the PINN generalization error, providing rigorous justificationfor the use of PINNs in this context. The abstract framework is illustratedwith examples of four prototypical linear PDEs. Numerical experiments,validating the proposed theory, are also presented.

Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs