Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many differ...
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| Published in | Nonlinearity Vol. 35; no. 1; pp. 278 - 310 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
IOP Publishing
06.01.2022
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0951-7715 1361-6544 |
| DOI | 10.1088/1361-6544/ac337f |
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| Abstract | In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future. |
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| AbstractList | In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future. |
| Author | E, Weinan Han, Jiequn Jentzen, Arnulf |
| Author_xml | – sequence: 1 givenname: Weinan orcidid: 0000-0003-0272-9500 surname: E fullname: E, Weinan organization: Princeton University Program in Applied and Computational Mathematics, United States of America – sequence: 2 givenname: Jiequn orcidid: 0000-0002-3553-7313 surname: Han fullname: Han, Jiequn organization: Flatiron Institute Center for Computational Mathematics, United States of America – sequence: 3 givenname: Arnulf orcidid: 0000-0002-9840-3339 surname: Jentzen fullname: Jentzen, Arnulf organization: The Chinese University of Hong Kong, Shenzhen School of Data Science and Shenzhen Research Institute of Big Data, China |
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| Title | Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning |
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