Deep splitting method for parabolic PDEs

Christian Beck; Sebastian Becker; Patrick Cheridito; Arnulf Jentzen; Ariel Neufeld

doi:10.1137/19M1297919

Deep splitting method for parabolic PDEs

Christian Beck^*, Sebastian Becker^*, Patrick Cheridito^*, Arnulf Jentzen^*, Ariel Neufeld^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

56 Citations (Scopus)

Abstract

In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

Original language	English
Pages (from-to)	A3135-A3154
Journal	SIAM Journal of Scientific Computing
Volume	43
Issue number	5
DOIs	https://doi.org/10.1137/19M1297919
Publication status	Published - 2021
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics

ASJC Scopus Subject Areas

Computational Mathematics
Applied Mathematics

Keywords

Deep learning
Neural networks
Nonlinear partial differential equations
Splitting-up method

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10.1137/19M1297919

Cite this

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abstract = "In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.",

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T1 - Deep splitting method for parabolic PDEs

AU - Beck, Christian

AU - Becker, Sebastian

AU - Cheridito, Patrick

AU - Jentzen, Arnulf

AU - Neufeld, Ariel

PY - 2021

Y1 - 2021

N2 - In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

AB - In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high dimensional PDEs. We test the method on different examples from physics, stochastic control, and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

KW - Deep learning

KW - Neural networks

KW - Nonlinear partial differential equations

KW - Splitting-up method

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SP - A3135-A3154

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

IS - 5

ER -

Deep splitting method for parabolic PDEs

Abstract

Bibliographical note

ASJC Scopus Subject Areas

Keywords

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