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

被引：19

作者：

Eigel, Martin ^{[1
]}

Schneider, Reinhold ^{[2
]}

Trunschke, Philipp ^{[2
]}

Wolf, Sebastian ^{[1
,2
]}

机构：

[1] Weierstrass Inst, Mohrenstr 39, D-10117 Berlin, Germany

[2] TU Berlin, Str 17 Juni 136, D-10623 Berlin, Germany

来源：

ADVANCES IN COMPUTATIONAL MATHEMATICS | 2019年 / 45卷 / 5-6期

关键词：

Machine learning; Uncertainty quantification; Partial differential equations; Statistical learning; Tree tensor networks; STOCHASTIC COLLOCATION METHOD; UNIFORM-CONVERGENCE; APPROXIMATION;

D O I：

10.1007/s10444-019-09723-8

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：2503 / 2532

页数：30

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