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.

中文翻译：

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估计物理信息神经网络 (PINNs) 近似 PDE 的泛化误差 II：一类逆问题

物理信息神经网络 (PINN) 最近已非常成功地应用于有效逼近 PDE 的逆问题。我们专注于一类特定的逆问题，即所谓的数据同化或独特的延续问题，并证明了对近似它们的 PINN 的泛化误差的严格估计。提出了一个抽象框架，并采用了对潜在逆问题的条件稳定性估计来推导出对 PINN 泛化误差的估计，为在这种情况下使用 PINN 提供了严格的理由。抽象框架用四个原型线性偏微分方程的例子来说明。还介绍了验证所提出的理论的数值实验。