Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^2(\mathbb{R}^d)$

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• Christoph Schwab (ETH Zürich)
• Tuesday 16 November 2021, 12:00-12:30
• Seminar Room 1, Newton Institute.

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MDLW03 - Deep learning and partial differential equations

For artificial deep neural networks, we prove expression rates for analytic functions $f:\mathbb{R}^{d \to \mathbb{R}$ in $L}2(\mathbb{R}^{d)$ where the dimension $d$ could be infinite, and where $L}2$ is with respect to gaussian measure.  We consider $\mbox{ReLU}$ and $\mbox{ReLU}^{k$ activations for integer $k\geq 2$.  In the infinite-dimensional case, under suitable smoothness and sparsity assumptions on $f:\mathbb{R}}{\mathbb{N}}\to \mathbb{R}$, with $\gamma_\infty$ denoting an infinite (Gaussian) product measure on $(\mathbb{R}^{}, {\mathcal B}(\mathbb{R}}{\mathbb{N}}))$, we prove dimension-independent  DNN expression rate bounds in the norm $L^2(\mathbb{R}{\mathbb{N}} , \gamma_\infty)$. The DNN expression rates are not subject to the CoD, and depend on summability of Wiener-Hermite expansion coefficients of $f$. Sufficient conditions are quantified holomorphy of (an analytic continuation of) the map $f$ on a product of strips in the complex domain. As application, we prove DNN expression rate bounds of deep $\mbox{ReLU}$-NNs for response surfaces of elliptic PDEs with log-gaussian random field inputs. (joint work with Jakob Zech, University of Heidelberg, Germany)

This talk is part of the Isaac Newton Institute Seminar Series series.

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