Zobrazeno 1 - 10
of 40
pro vyhledávání: '"Kuckuck, Benno"'
Autor:
Gonon, Lukas, Jentzen, Arnulf, Kuckuck, Benno, Liang, Siyu, Riekert, Adrian, von Wurstemberger, Philippe
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively studied. One
Externí odkaz:
http://arxiv.org/abs/2408.13222
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that the number
Externí odkaz:
http://arxiv.org/abs/2406.10876
This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including different artificial neural network (ANN) architectures (such as full
Externí odkaz:
http://arxiv.org/abs/2310.20360
Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations which appea
Externí odkaz:
http://arxiv.org/abs/2309.13722
Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for cer
Externí odkaz:
http://arxiv.org/abs/2205.03672
Autor:
Beneventano, Pierfrancesco, Cheridito, Patrick, Graeber, Robin, Jentzen, Arnulf, Kuckuck, Benno
The purpose of this article is to develop machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of dimensionality in
Externí odkaz:
http://arxiv.org/abs/2112.14523
Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general time horizon
Externí odkaz:
http://arxiv.org/abs/2110.08297
Publikováno v:
Discrete Contin. Dyn. Syst. Ser. B 28 (2023), no. 6, 3697-3746
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been pro
Externí odkaz:
http://arxiv.org/abs/2012.12348
Publikováno v:
Discrete Contin. Dyn. Syst. Ser. B 27 (2022), no. 7, 3707-3724
In the recent article [A. Jentzen, B. Kuckuck, T. M\"uller-Gronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coeffici
Externí odkaz:
http://arxiv.org/abs/2001.03472
Publikováno v:
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 25 (2022), no. 2, 2150020, 77 pp
Deep learning algorithms have been applied very successfully in recent years to a range of problems out of reach for classical solution paradigms. Nevertheless, there is no completely rigorous mathematical error and convergence analysis which explain
Externí odkaz:
http://arxiv.org/abs/1910.00121