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Deep Learning in High Dimension

by Ch. Schwab and J. Zech

(Report number 2017-57)

Abstract
We estimate the expressive power of a class of deep Neural Networks (DNNs for short) on a class of countably-parametric maps \(u:U\to \mathbb{R}\) on the parameter domain \(U=[-1,1]^{\mathbb{N}}\). Such maps arise for example as response surfaces of parametric PDEs with distributed uncertain inputs, i.e., input data from function spaces. Equipping these spaces with suitable bases, instances of uncertain inputs become sequences of (coefficient) parameters of representations in these bases. Dimension-independent approximation rates of generalized polynomial chaos (gpc for short) approximations of countably-parametric maps \(u:U\to \mathbb{R}\) depend only on the degree of sparsity of the gpc expansion of \(u\) as quantified by the summability exponent of the sequence of their gpc expansion coefficients: for parametric maps which are \(p\)-sparse with some \(0 < p < 1\), we show that a certain architecture of DNNs afford the same convergence rates in terms of \(N\), the total number of units in the DNN. So-called \(({\boldsymbol b},\varepsilon)\)-holomorphic maps \(u\) with \({\boldsymbol b}\in\ell^p\) for some \(p\in(0,1)\) arise in a number of applications from computational uncertainty quantification. For this class of functions, up to logarithmic factors we prove the dimension independent approximation rate \(s = 1/p-1\) in terms of the total number \(N\) of units in the DNN. This shows that the DNN architectures can overcome the curse of dimensionality when expressing possibly infinite-parametric, real-valued maps with a certain sparsity. Examples of such maps comprise response maps of parametric and stochastic PDEs models with distributed uncertain input data.

Keywords:

@Techreport{SZ17_753,
  author = {Ch. Schwab and J. Zech},
  title = {Deep Learning in High Dimension},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-57},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-57.pdf },
  year = {2017}
}

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