Estimates on the generalization error of physics-informed neural networks for approximating PDEs. (14th January 2022)

• Record Type: Journal Article
• Authors: Mishra, Siddhartha; Molinaro, Roberto
• Is Part Of: IMA journal of numerical analysis. Volume 43:Number 1(2023)
• Page Start: 1
• Page End: 43
• Publication Date: 2022-01-14
• Subjects: PDEs -- neural networks -- PINNs -- stability estimates -- generalization error -- quadrature; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drab093
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: Physics-informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of partial differential equations (PDEs). We provide upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.