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Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

by C. Marcati and Ch. Schwab

(Report number 2021-42)

Abstract
We prove exponential expression rates of deep operator networks (deep ONets) between infinite-di\-men\-sion\-al spaces that emulate the coefficient-to-solution map of linear, elliptic, second-order, divergence-form partial differential equations (PDEs). In particular, we consider problems set in $d$-dimensional periodic domains, $d=1, 2, \dots$, with analytic right-hand sides and coefficients. Our analysis covers diffusion-reaction problems, parametric diffusion equations, and certain elliptic systems such as linear isotropic elastostatics in heterogeneous materials. In the constructive proofs, we leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the branch and trunk ONet architecture of Chen and Chen \cite{ChenChen1993} and of Lu et al.~\cite{lu2020deeponet}, we construct deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the $H^1$ norm, uniformly over the PDE coefficient set. The ONets have size $\mathcal{O}(\left|\log(\varepsilon)\right|^\kappa)$, where $\varepsilon>0$ is the approximation accuracy, for some $\kappa>0$ depending on the physical space dimension.

Keywords: Operator networks, deep neural networks, exponential convergence, elliptic PDEs

@Techreport{MS21_984,
  author = {C. Marcati and Ch. Schwab},
  title = {Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-42},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-42.pdf },
  year = {2021}
}

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