26 citations · 96 across the 22 of their papers we have counts for
23 papers
Asymptotic stability properties and a priori bounds for Adam and other gradient descent optimization methods
Steffen Dereich, Robin Graeber, Arnulf Jentzen +1
Gradient descent (GD) based optimization methods are these days the standard tools to train deep neural networks in artificial intelligence systems. In optimization procedures in d…
PADAM: Parallel averaged Adam reduces the error for stochastic optimization in scientific machine learning
Arnulf Jentzen, Julian Kranz, Adrian Riekert
Averaging techniques such as Ruppert--Polyak averaging and exponential movering averaging (EMA) are powerful approaches to accelerate optimization procedures of stochastic gradient…
Sharp higher order convergence rates for the Adam optimizer
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
Gradient descent based optimization methods are the methods of choice to train deep neural networks in machine learning. Beyond the standard gradient descent method, also suitable…
Non-convergence to the optimal risk for Adam and stochastic gradient descent optimization in the training of deep neural networks
Thang Do, Arnulf Jentzen, Adrian Riekert
Despite the omnipresent use of stochastic gradient descent (SGD) optimization methods in the training of deep neural networks (DNNs), it remains, in basically all practically relev…
Averaged Adam accelerates stochastic optimization in the training of deep neural network approximations for partial differential equation and optimal control problems
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
Deep learning methods - usually consisting of a class of deep neural networks (DNNs) trained by a stochastic gradient descent (SGD) optimization method - are nowadays omnipresent i…
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…
Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to…
Non-convergence to global minimizers for Adam and stochastic gradient descent optimization and constructions of local minimizers in the training of artificial neural networks
Arnulf Jentzen, Adrian Riekert
Stochastic gradient descent (SGD) optimization methods such as the plain vanilla SGD method and the popular Adam optimizer are nowadays the method of choice in the training of arti…
A proof of the corrected Sister Beiter cyclotomic coefficient conjecture inspired by Zhao and Zhang
Branko Juran, Pieter Moree, Adrian Riekert +2
The largest coefficient (in absolute value) of a cyclotomic polynomial is called its height . In case is a fixed prime it turns out that as and range over a…
Deep neural network approximation of composite functions without the curse of dimensionality
Adrian Riekert
In this article we identify a general class of high-dimensional continuous functions that can be approximated by deep neural networks (DNNs) with the rectified linear unit (ReLU) a…
Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations
Arnulf Jentzen, Adrian Riekert, Philippe von Wurstemberger
In this article we propose a new deep learning approach to approximate operators related to parametric partial differential equations (PDEs). In particular, we introduce a new stra…
Convergence to good non-optimal critical points in the training of neural networks: Gradient descent optimization with one random initialization overcomes all bad non-global local minima with high probability
Shokhrukh Ibragimov, Arnulf Jentzen, Adrian Riekert
Gradient descent (GD) methods for the training of artificial neural networks (ANNs) belong nowadays to the most heavily employed computational schemes in the digital world. Despite…
Normalized gradient flow optimization in the training of ReLU artificial neural networks
Simon Eberle, Arnulf Jentzen, Adrian Riekert +1
The training of artificial neural networks (ANNs) is nowadays a highly relevant algorithmic procedure with many applications in science and industry. Roughly speaking, ANNs can be…
On the existence of infinitely many realization functions of non-global local minima for artificial neural networks with ReLU activation
Shokhrukh Ibragimov, Arnulf Jentzen, Timo Kröger +1
Gradient descent (GD) type optimization schemes are standard tools for training fully connected feedforward artificial neural networks (ANNs) with rectified linear unit (ReLU) acti…
On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks
Arnulf Jentzen, Adrian Riekert
In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization me…
Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions
Martin Hutzenthaler, Arnulf Jentzen, Katharina Pohl +2
In many numerical simulations stochastic gradient descent (SGD) type optimization methods perform very effectively in the training of deep neural networks (DNNs) but till this day…
Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation
Simon Eberle, Arnulf Jentzen, Adrian Riekert +1
The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially…
A proof of convergence for the gradient descent optimization method with random initializations in the training of neural networks with ReLU activation for piecewise linear target functions
Arnulf Jentzen, Adrian Riekert
Gradient descent (GD) type optimization methods are the standard instrument to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Despite the gre…
Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation
Arnulf Jentzen, Adrian Riekert
Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can b…
A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions
Arnulf Jentzen, Adrian Riekert
In this article we study the stochastic gradient descent (SGD) optimization method in the training of fully-connected feedforward artificial neural networks with ReLU activation. T…
A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions
Patrick Cheridito, Arnulf Jentzen, Adrian Riekert +1
Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number of numerical simulations i…
Convergence Rates for Empirical Measures of Markov Chains in Dual and Wasserstein Distances
Adrian Riekert
We consider a Markov chain on with invariant measure . We are interested in the rate of convergence of the empirical measures towards the invariant measure with r…
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…