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Strong overall error analysis for the training of artificial neural networks via random initializations
Arnulf Jentzen, Adrian Riekert
Although deep learning based approximation algorithms have been applied very successfully to numerous problems, at the moment the reasons for their performance are not entirely und…
An overview on deep learning-based approximation methods for partial differential equations
Christian Beck, Martin Hutzenthaler, Arnulf Jentzen +1
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-b…
High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations
Pierfrancesco Beneventano, Patrick Cheridito, Arnulf Jentzen +1
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of di…
Deep learning based numerical approximation algorithms for stochastic partial differential equations
Christian Beck, Sebastian Becker, Patrick Cheridito +2
In this article, we introduce a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realiz…
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of h…
Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations
Christian Beck, Arnulf Jentzen, Thomas Kruse
The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approxim…