Deep neural network expression of posterior expectations in Bayesian PDE inversion*The main part of the paper was written while LH was at the Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH–8092 Zürich, Switzerland, and during the postdoctoral stay of JZ to the Department of Aeronautics and Astronautics, MIT, 02139 Cambridge, MA, USA. JZ is supported by the Swiss National Science Foundation under Early Postdoc.Mobility Fellowship 184530. CS acknowledges stimulating discussions at the RICAM WS on Optimization under uncertainty in November 2019 at RICAM, Linz, Austria, and at the WIAS WS on Deep Learning for PDEs at the Weierstrass Institute Berlin, Germany, 2–6 December 2019. (3rd December 2020) - Herrmann, Lukas; Schwab, Christoph; Zech, Jakob
Abstract: For Bayesian inverse problems with input-to-response maps given by well-posed partial differential equations 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 rate bounds for this data-to-QoI map by deep neural networks with rectified linear unit activation function, which are uniform with respect to the data δ taking values in a compact subset of R K . Similar convergence rates are verified for polynomial and rational approximations of the data-to-QoI map. We discuss the extension to other activation functions, and to mere Lipschitz continuity of the data-to-QoI map.