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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| Published in | Advances in computational mathematics Vol. 45; no. 5-6; pp. 2503 - 2532 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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New York
Springer US
01.12.2019
Springer Nature B.V |
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| Abstract | 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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| AbstractList | 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. |
| Author | Schneider, Reinhold Trunschke, Philipp Eigel, Martin Wolf, Sebastian |
| Author_xml | – sequence: 1 givenname: Martin surname: Eigel fullname: Eigel, Martin organization: Weierstrass Institute – sequence: 2 givenname: Reinhold surname: Schneider fullname: Schneider, Reinhold organization: TU Berlin – sequence: 3 givenname: Philipp orcidid: 0000-0002-2995-126X surname: Trunschke fullname: Trunschke, Philipp email: ptrunschke@mail.tu-berlin.de organization: TU Berlin – sequence: 4 givenname: Sebastian surname: Wolf fullname: Wolf, Sebastian organization: Weierstrass Institute, TU Berlin |
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| Keywords | 62J02 35R60 Partial differential equations Statistical learning Uncertainty quantification 15A69 Tree tensor networks 65C20 41A63 Machine learning 65D15 68Q32 65N22 65N30 |
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| Snippet | A statistical learning approach for high-dimensional parametric PDEs related to uncertainty quantification is derived. The method is based on the minimization... |
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| SubjectTerms | 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 |
| Title | Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations |
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