Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. (17th June 2021)

• Record Type: Journal Article
• Authors: Mishra, Siddhartha; Molinaro, Roberto
• Is Part Of: IMA journal of numerical analysis. Volume 42:Number 2(2022)
• Page Start: 981
• Page End: 1022
• Publication Date: 2021-06-17
• Subjects: PDEs -- inverse problems -- data assimilation -- neural networks -- PINNs -- generalization error -- quadrature -- conditional stability; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drab032
• Languages: English
• ISSNs: 0272-4979

Abstract

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.