Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
Numerical Analysis
2023-12-07 v3 Machine Learning
Numerical Analysis
Analysis of PDEs
Abstract
Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous 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.
Keywords
Cite
@article{arxiv.2006.16144,
title = {Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs},
author = {Siddhartha Mishra and Roberto Molinaro},
journal= {arXiv preprint arXiv:2006.16144},
year = {2023}
}