Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Uniform error estimates for artificial neural network approximations for heat equations

by L. Gonon and Ph. Grohs and A. Jentzen and D. Kofler and D. Šiška

(Report number 2019-61)

Abstract
Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature which examine the approximation capabilities of such deep learning based approximation algorithms for PDEs. These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs. In these mathematical results from the scientific literature usually the error between the solution of the PDE and the approximating ANN is measured in the \(L^p\)-sense with respect to some \(p \in [1,\infty)\) and some probability measure. In many applications it is, however, also important to control the error in a uniform \(L^\infty\)-sense. The key contribution of the main result of this article is to develop the techniques to obtain error estimates between solutions of PDEs and approximating ANNs in the uniform \(L^\infty\)-sense. In particular, we prove that the number of parameters of an ANN to uniformly approximate the classical solution of the heat equation in a region \([a,b]^d\) for a fixed time point \(T \in (0,\infty)\) grows at most polynomially in the dimension \(d \in \mathbb{N}\) and the reciprocal of the approximation precision \(\varepsilon > 0\). This shows that ANNs can overcome the curse of dimensionality in the numerical approximation of the heat equation when the error is measured in the uniform \(L^\infty\)-norm.

Keywords:

@Techreport{GGJKŠ19_865,
  author = {L. Gonon and Ph. Grohs and A. Jentzen and D. Kofler and D. Šiška},
  title = {Uniform error estimates for artificial neural network approximations for heat equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-61},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-61.pdf },
  year = {2019}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).