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Deep ReLU Neural Network Expression Rates for Data-to-QoI Maps in Bayesian PDE Inversion

by L. Herrmann and Ch. Schwab and J. Zech

(Report number 2020-02)

Abstract
For 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 \(\delta\) to the Bayesian estimate for an unknown quantity of interest (QoI). We prove exponential expression rate bounds for this data-to-QoI map by deep neural networks with rectified linear unit (ReLU) activation function, which are uniform with respect to the data \(\delta\) taking values in a compact subset of \(\mathbb{R}^K\). Similar convergence rates are verified for polynomial and rational approximations of the data-to-QoI map.

Keywords: Deep ReLU neural networks, Bayesian inverse problems, approximation rates, exponential convergence, Uncertainty Quantification

@Techreport{HSZ20_875,
  author = {L. Herrmann and Ch. Schwab and J. Zech},
  title = {Deep ReLU Neural Network Expression Rates for Data-to-QoI Maps in Bayesian PDE Inversion},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-02.pdf },
  year = {2020}
}

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