Deep neural network expression of posterior expectations in Bayesian PDE inversion
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...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
3 December 2020
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| In: |
Inverse problems
Year: 2020, Volume: 36, Issue: 12 |
| ISSN: | 1361-6420 |
| DOI: | 10.1088/1361-6420/abaf64 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1088/1361-6420/abaf64 Verlag, lizenzpflichtig, Volltext: https://iopscience.iop.org/article/10.1088/1361-6420/abaf64 
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| Author Notes: | Lukas Herrmann, Christoph Schwab and Jakob Zech |
MARC
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| 520 | |a 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 . 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. | ||
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