Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory

1 code implementation • 31 Oct 2023 • Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger

This book aims to provide an introduction to the topic of deep learning algorithms.

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 $L^p$-sense

no code implementations • 24 Sep 2023 • Julia Ackermann, Arnulf Jentzen, Thomas Kruse, Benno Kuckuck, Joshua Lee Padgett

Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed.

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

no code implementations • 7 May 2022 • Victor Boussange, Sebastian Becker, Arnulf Jentzen, Benno Kuckuck, Loïc Pellissier

We evaluate the performance of the two methods on five different PDEs arising in physics and biology.

An overview on deep learning-based approximation methods for partial differential equations

no code implementations • 22 Dec 2020 • Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs).

Full error analysis for the training of deep neural networks

no code implementations • 30 Sep 2019 • Christan Beck, Arnulf Jentzen, Benno Kuckuck

In this work we estimate for a certain deep learning algorithm each of these three errors and combine these three error estimates to obtain an overall error analysis for the deep learning algorithm under consideration.

Strong error analysis for stochastic gradient descent optimization algorithms

no code implementations • 29 Jan 2018 • Arnulf Jentzen, Benno Kuckuck, Ariel Neufeld, Philippe von Wurstemberger

Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications.

Numerical Analysis Probability