Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

A statistical learning approach for high-dimensional parametric PDEs related to uncertainty quantification is derived. The method is based on the minimization of an empirical risk on a selected model class, and it is shown to be applicable to a broad range of problems. A general unified convergence...

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Bibliographic Details
Published inAdvances in computational mathematics Vol. 45; no. 5-6; pp. 2503 - 2532
Main Authors Eigel, Martin, Schneider, Reinhold, Trunschke, Philipp, Wolf, Sebastian
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2019
Springer Nature B.V
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Summary:A statistical learning approach for high-dimensional parametric PDEs related to uncertainty quantification is derived. The method is based on the minimization of an empirical risk on a selected model class, and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-019-09723-8