Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs

作者：Siddhartha Kumar Mishra, Roberto Molinaro · 发表于：IMA Journal of Numerical Analysis · 年份：2021 · DOI：10.1093/imanum/drab032 · 被引用次数：333 · 研究领域：Model Reduction and Neural Networks、Magnetic Properties and Applications、Numerical methods in inverse problems

Abstract Physics-informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for partial differential equations (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.