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Deep Operator Network Approximation Rates for Lipschitz Operators
(2023)SAM Research ReportWe establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or Hölder) continuous maps \(\mathcal G:\mathcal X\to\mathcal Y\) between (subsets of) separable Hilbert spaces \(\mathcal X\), \(\mathcal Y\). The DON architecture considered uses linear encoders \(\mathcal E\) and decoders \(\mathcal D\) via (biorthogonal) Riesz bases of \(\mathcal X\), \(\mathcal Y\), and an ...Report -
Deep learning in high dimension: ReLU network Expression Rates for Bayesian PDE inversion
(2020)SAM Research ReportWe establish dimension independent expression rates by deep ReLU networks for so-called (b,ε,X)-holomorphic functions. These are mappings from [−1,1]N→X, with X being a Banach space, that admit analytic extensions to certain polyellipses in each of the input variables. The significance of this function class has been established in previous works, where it was shown that functions of this type occur widely in uncertainty quantification ...Report -
Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs
(2022)SAM Research ReportWe establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments ...Report -
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Electromagnetic Wave Scattering by Random Surfaces: Shape Holomorphy
(2016)Research reports / Seminar for Applied MathematicsReport -
Multilevel Domain Uncertainty Quantification in Computational Electromagnetics
(2022)SAM Research ReportWe continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), {\bf 8}:301--341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback ...Report -
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Deep ReLU Neural Network Expression Rates for Data-to-QoI Maps in Bayesian PDE Inversion
(2020)SAM Research ReportFor Bayesian inverse problems with input-to-response maps given by well-posed partial differential equations (PDEs) and subject to uncertain parametric or function space input, we establish (under rather weak conditions on the ``forward'', input-to-response maps) the parametric holomorphy of the data-to-QoI map relating observation data 𝛿� to the Bayesian estimate for an unknown quantity of interest (QoI). We prove exponential expression ...Report -
Neural and gpc operator surrogates: construction and expression rate bounds
(2022)SAM Research ReportApproximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs ...Report