Research reports

Years: 2025 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

Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes

by C. Marcati and Ch. Schwab

(Report number 2024-23)

Abstract
We study the approximation rates of a class of deep neural network approximations of operators, which arise as data-to-solution maps ${G}^\dagger$ of linear elliptic partial differential equations (PDEs), and act between pairs $X,Y$ of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators ${G}$ with the structure ${G} = {R} \circ {A} \circ {E}$, with linear encoders ${E}$ and decoders ${R}$. The constructive proofs are via a recurrent NN structure obtained by unrolling exponentially convergent, self-consistent iterations. We bound the operator approximation error with respect to the linear Kolmogorov $N$-widths of the data and solution sets and in terms of the size of the approximation network. We derive expression rates for full neural operators for elliptic PDEs set in $d$-dimensional polytopes, with $d\in\{2,3\}$, and with Dirichlet-, Neumann- or mixed boundary conditions. Exploiting the regularity of the solutions sets of elliptic PDEs in polytopes, we show algebraic rates of expression for problems with data with finite regularity, and exponential operator expression rates for analytic data.

Keywords: Neural Operators, Approximation Theory, Deep Neural Networks, Elliptic PDEs, Kolmogorov $N$-widths

@Techreport{MS24_1105,
  author = {C. Marcati and Ch. Schwab},
  title = {Expression Rates of Neural Operators
for Linear Elliptic PDEs in Polytopes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-23.pdf },
  year = {2024}
}

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).