Convergence Rates for a Deep Learning Algorithm for Semilinear PDEs

39 Pages Posted: 16 Feb 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

Maximilian Würschmidt

University of Trier

Date Written: December 9, 2021

Abstract

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

Maximilian Würschmidt

University of Trier ( email )

15, Universitaetsring
Trier, 54286
Germany

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