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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 construct deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order PDEs. In particular, we consider problems set in $d$-dimensional periodic domains, $d=1, 2, \dots$, and with analytic right-hand sides and coefficients. Our analysis covers diffusion-reaction problems, parametric diffusion equations, and elliptic systems such as linear isotropic elastostatics in heterogeneous materials. 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 ONet branch and trunk construction of [Chen and Chen, 1993] and of [Lu et al., 2021], we show the existence of deep ONets which emulate the coefficient-to-solution map to accuracy $\varepsilon>0$ in the $H^1$ norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size $\mathcal{O}(\left|\log(\varepsilon)\right|^\kappa)$ 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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