88 citations · 204 across the 14 of their papers we have counts for
17 papers
On the gradient conjecture at infinity: Finite spherical length for gradient trajectories of definable potentials
Thang Do, Arnulf Jentzen, Benno Kuckuck
We prove that for every open the spherical projection of every gradient trajectory , , of a definable -potential $F \colon U \to \m…
Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients
Benno Kuckuck
The -- theorem, recently established by Cohen and Shusterman, says that if two groups of type map onto a common quotient of type $\mathrm{F}…
On the logical skills of large language models: evaluations using arbitrarily complex first-order logic problems
Shokhrukh Ibragimov, Arnulf Jentzen, Benno Kuckuck
We present a method of generating first-order logic statements whose complexity can be controlled along multiple dimensions. We use this method to automatically create several data…
An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning
Lukas Gonon, Arnulf Jentzen, Benno Kuckuck +3
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 meth…
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations
Julia Ackermann, Arnulf Jentzen, Benno Kuckuck +1
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer…
Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory
Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger
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…
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the -sense
Julia Ackermann, Arnulf Jentzen, Thomas Kruse +2
Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have gener…
Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions
Victor Boussange, Sebastian Becker, Arnulf Jentzen +2
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…
Deep neural network approximation theory for high-dimensional functions
Pierfrancesco Beneventano, Patrick Cheridito, Robin Graeber +2
The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DN…
Strong -error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck +1
Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semi…
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…
Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives
Arnulf Jentzen, Benno Kuckuck, Thomas Müller-Gronbach +1
In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions to every additive noise driv…
Full error analysis for the training of deep neural networks
Christan Beck, Arnulf Jentzen, Benno Kuckuck
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 complet…
On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values
Arnulf Jentzen, Benno Kuckuck, Thomas Müller-Gronbach +1
Recently in [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab. 43, 2 (2015), 468--527] and [A. Jentzen, T. Müller-Gronbach, and L. Yaroslavtseva, Commun. Math. Sci. 14, 6 (2…
Strong error analysis for stochastic gradient descent optimization algorithms
Arnulf Jentzen, Benno Kuckuck, Ariel Neufeld +1
Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analy…
The Magnus property for direct products
B. Klopsch, B. Kuckuck
A group G is said to have the Magnus property if the following holds: whenever two elements x,y have the same normal closure, then x is conjugate to y or its inverse. We prove: Let…
Subdirect products of groups and the n-(n+1)-(n+2) Conjecture
Benno Kuckuck
We analyse the subgroup structure of direct products of groups. Earlier work on this topic has revealed that higher finiteness properties play a crucial role in determining which g…