Convergence Rates for a Deep Learning Algorithm for Semilinear PDEs

42 Pages Posted: 16 Feb 2022 Last revised: 21 Jun 2022

See all articles by Christoph Belak

Christoph Belak

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften

Oliver Hager

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften

Charlotte Reimers

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften

Lotte Schnell

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften

Frank Thomas Seifried

University of Trier

Maximilian Würschmidt

University of Trier

Date Written: December 9, 2021

Abstract

We derive convergence rates for a deep solver for semilinear partial differential equations which is based on a Feynman-Kac representation in terms of a forward-backward stochastic differential equation and a discretization in time. We show that the error of the deep solver is bounded in terms of its loss functional, hence yielding a direct measure to judge the quality in numerical applications, and that the loss functional converges sufficiently fast to zero to guarantee that the approximation error vanishes in the limit. As a consequence of these results, we show that the deep solver has a strong convergence rate of order 1/2.

Keywords: semilinear PDE, forward-backward SDE, deep solver, strong convergence rate

Suggested Citation

Belak, Christoph and Hager, Oliver and Reimers, Charlotte and Schnell, Lotte and Seifried, Frank Thomas and Würschmidt, Maximilian, Convergence Rates for a Deep Learning Algorithm for Semilinear PDEs (December 9, 2021). Available at SSRN: https://ssrn.com/abstract=3981933 or http://dx.doi.org/10.2139/ssrn.3981933

Christoph Belak (Contact Author)

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften ( email )

Institut fur Mathematik, Sekr. MA 7-1
Strasse des 17. Juni 136
Berlin, 10623
Germany

Oliver Hager

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften ( email )

Institut fur Mathematik, Sekr. MA 6-1
Strasse des 17. Juni 136
Berlin, 10623
Germany

Charlotte Reimers

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften ( email )

Institut fur Mathematik, Sekr. MA 6-1
Strasse des 17. Juni 136
Berlin, 10623
Germany

Lotte Schnell

Technische Universität Berlin (TU Berlin) - Fakultat II - Mathematik und Naturwissenschaften ( email )

Institut fur Mathematik, Sekr. MA 6-1
Strasse des 17. Juni 136
Berlin, 10623
Germany

Frank Thomas Seifried

University of Trier ( email )

Department IV - Mathematics
Universitätsring 19
Trier, 54296
Germany

HOME PAGE: http://sites.google.com/site/seifriedfinance/

Maximilian Würschmidt

University of Trier ( email )

15, Universitaetsring
Trier, 54286
Germany

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