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

• Published in: Advances in computational mathematics Vol. 45; no. 5-6; pp. 2503 - 2532
• Main Authors: Eigel, Martin, Schneider, Reinhold, Trunschke, Philipp, Wolf, Sebastian
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Computer simulation; Empirical analysis; Machine learning; Mathematical analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Model reduction of parametrized Systems; Numerical analysis; Partial differential equations; Tensors; Visualization; 62J02; 35R60; Partial differential equations; Statistical learning; Uncertainty quantification; 15A69; Tree tensor networks; 65C20; 41A63; Machine learning; 65D15; 68Q32; 65N22; 65N30
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09723-8

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.