Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

arXiv:2112.08125v3 Announce Type: replace-cross Abstract: We construct and analyze approximation rates of 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 partial differential equations. 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 linear, elliptic second order divergence-form PDEs as, e.g., 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 a desired accuracy in the $H^1$ norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size $\mathcal{O}(\left|\log(\varepsilon)\right|^\kappa)$, where $\varepsilon>0$ is the approximation accuracy, for some $\kappa>0$ depending on the physical space dimension.

Abstract:We construct and analyze approximation rates of 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 partial differential equations. 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 linear, elliptic second order divergence-form PDEs as, e.g., 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 a desired accuracy in the $H^1$ norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size $\mathcal{O}(\left|\log(\varepsilon)\right|^\kappa)$, where $\varepsilon>0$ is the approximation accuracy, for some $\kappa>0$ depending on the physical space dimension.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)

arXiv:2112.08125 [math.NA]

(or arXiv:2112.08125v3 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2112.08125 Focus to learn more arXiv-issued DOI via DataCite

SIAM J. Numer. Anal.61(2023), no.3, 1513-1545

https://doi.org/10.1137/21M1465718 Focus to learn more DOI(s) linking to related resources

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Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

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