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.

翻译：近年来，物理信息神经网络（PINNs）在高效近似偏微分方程反问题中取得了非常成功的应用。我们聚焦于特定的一类反问题——即数据同化或唯一延拓问题，并给出了PINNs近似该类问题时泛化误差的严格估计。本文提出了一个抽象框架，并利用底层反问题的条件稳定性估计推导了PINN泛化误差的估计式，从而为PINNs在此类问题中的应用提供了严格的理论依据。通过四个典型线性偏微分方程的实例对该抽象框架进行了说明，同时给出了验证所提理论的数值实验。