论文标题
估计物理知情神经网络（PINN）的概述，以近似PDE的一类反向问题
Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs
论文作者
论文摘要
物理知情的神经网络（PINN）最近非常成功地应用了PDE的逆问题。我们专注于一类特定的反问题，所谓的数据同化或唯一的延续问题，并证明对近似它们的PINN的概括误差进行了严格的估计。提出了一个抽象框架，并采用了基础反问题的有条件稳定性估计来得出对PINN概括误差的估计，从而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PDE的示例进行了说明。还提出了验证所提出的理论的数值实验。
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.
