Arnulf Jentzen

research

∙ 02/28/2023

On the existence of minimizers in shallow residual ReLU neural network optimization landscapes

Many mathematical convergence results for gradient descent (GD) based al...

0 Steffen Dereich, et al. ∙

research

∙ 02/07/2023

Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations

In this article we propose a new deep learning approach to solve paramet...

0 Arnulf Jentzen, et al. ∙

research

∙ 01/19/2023

The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality

In this article we study high-dimensional approximation capacities of sh...

0 Lukas Gonon, et al. ∙

research

∙ 10/24/2022

An efficient Monte Carlo scheme for Zakai equations

In this paper we develop a numerical method for efficiently approximatin...

0 Christian Beck, et al. ∙

research

∙ 08/03/2022

Gradient descent provably escapes saddle points in the training of shallow ReLU networks

Dynamical systems theory has recently been applied in optimization to pr...

0 Patrick Cheridito, et al. ∙

research

∙ 07/13/2022

Normalized gradient flow optimization in the training of ReLU artificial neural networks

The training of artificial neural networks (ANNs) is nowadays a highly r...

0 Simon Eberle, et al. ∙

research

∙ 06/27/2022

On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector

It is an elementary fact in the scientific literature that the Lipschitz...

0 Arnulf Jentzen, et al. ∙

research

∙ 05/07/2022

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

Nonlinear partial differential equations (PDEs) are used to model dynami...

0 Victor Boussange, et al. ∙

research

∙ 02/06/2022

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

In financial engineering, prices of financial products are computed appr...

0 Sebastian Becker, et al. ∙

research

∙ 12/29/2021

Deep neural network approximation theory for high-dimensional functions

The purpose of this article is to develop machinery to study the capacit...

0 Pierfrancesco Beneventano, et al. ∙

research

∙ 12/17/2021

On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks

In this article we study fully-connected feedforward deep ReLU ANNs with...

0 Arnulf Jentzen, et al. ∙

research

∙ 12/13/2021

Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions

In many numerical simulations stochastic gradient descent (SGD) type opt...

0 Martin Hutzenthaler, et al. ∙

research

∙ 10/15/2021

Strong L^p-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations

Full-history recursive multilevel Picard (MLP) approximation schemes hav...

0 Martin Hutzenthaler, et al. ∙

research

∙ 08/24/2021

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

Backward stochastic differential equations (BSDEs) belong nowadays to th...

0 Martin Hutzenthaler, et al. ∙

research

∙ 08/18/2021

Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation

The training of artificial neural networks (ANNs) with rectified linear ...

0 Simon Eberle, et al. ∙

research

∙ 08/10/2021

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 tar

Gradient descent (GD) type optimization methods are the standard instrum...

0 Arnulf Jentzen, et al. ∙

research

∙ 07/09/2021

Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation

Gradient descent (GD) type optimization schemes are the standard methods...

0 Arnulf Jentzen, et al. ∙

research

∙ 04/01/2021

A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions

In this article we study the stochastic gradient descent (SGD) optimizat...

0 Arnulf Jentzen, et al. ∙

research

∙ 03/19/2021

Landscape analysis for shallow ReLU neural networks: complete classification of critical points for affine target functions

In this paper, we analyze the landscape of the true loss of a ReLU neura...

0 Patrick Cheridito, et al. ∙

research

∙ 03/07/2021

Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality

Artificial neural networks (ANNs) have become a very powerful tool in th...

0 Philipp Grohs, et al. ∙

research

∙ 03/03/2021

Full history recursive multilevel Picard approximations for ordinary differential equations with expectations

We consider ordinary differential equations (ODEs) which involve expecta...

0 Christian Beck, et al. ∙

research

∙ 02/23/2021

Convergence rates for gradient descent in the training of overparameterized artificial neural networks with biases

In recent years, artificial neural networks have developed into a powerf...

0 Arnulf Jentzen, et al. ∙

research

∙ 02/19/2021

A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions

Gradient descent optimization algorithms are the standard ingredients th...

0 Patrick Cheridito, et al. ∙

research

∙ 12/22/2020

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

It is one of the most challenging problems in applied mathematics to app...

0 Christian Beck, et al. ∙

research

∙ 12/15/2020

Strong overall error analysis for the training of artificial neural networks via random initializations

Although deep learning based approximation algorithms have been applied ...

0 Arnulf Jentzen, et al. ∙

research

∙ 12/08/2020

High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations

In this paper we develop a new machinery to study the capacity of artifi...

0 Pierfrancesco Beneventano, et al. ∙

research

∙ 12/02/2020

Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems

In this article we introduce and study a deep learning based approximati...

0 Christian Beck, et al. ∙

research

∙ 09/29/2020

Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations

The approximative calculation of iterated nested expectations is a recur...

0 Christian Beck, et al. ∙

research

∙ 09/05/2020

Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities

The recently introduced full-history recursive multilevel Picard (MLP) a...

0 Martin Hutzenthaler, et al. ∙

research

∙ 08/31/2020

Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

In recent years, tremendous progress has been made on numerical algorith...

0 Weinan E, et al. ∙

research

∙ 07/03/2020

Weak error analysis for stochastic gradient descent optimization algorithms

Stochastic gradient descent (SGD) type optimization schemes are fundamen...

0 Aritz Bercher, et al. ∙

research

∙ 06/12/2020

Non-convergence of stochastic gradient descent in the training of deep neural networks

Deep neural networks have successfully been trained in various applicati...

0 Patrick Cheridito, et al. ∙

research

∙ 06/03/2020

Space-time deep neural network approximations for high-dimensional partial differential equations

It is one of the most challenging issues in applied mathematics to appro...

0 Fabian Hornung, et al. ∙

research

∙ 05/20/2020

Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations

One of the most challenging issues in applied mathematics is to develop ...

0 Sebastian Becker, et al. ∙

research

∙ 03/03/2020

Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation

In spite of the accomplishments of deep learning based algorithms in num...

0 Arnulf Jentzen, et al. ∙

research

∙ 03/01/2020

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Recently, so-called full-history recursive multilevel Picard (MLP) appro...

0 Christian Beck, et al. ∙

research

∙ 12/09/2019

Efficient approximation of high-dimensional functions with deep neural networks

In this paper, we develop an approximation theory for deep neural networ...

0 Patrick Cheridito, et al. ∙

research

∙ 12/05/2019

Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities

Partial differential equations (PDEs) are a fundamental tool in the mode...

0 Martin Hutzenthaler, et al. ∙

research

∙ 11/20/2019

Uniform error estimates for artificial neural network approximations for heat equations

Recently, artificial neural networks (ANNs) in conjunction with stochast...

0 Lukas Gonon, et al. ∙

research

∙ 11/08/2019

Generalised multilevel Picard approximations

It is one of the most challenging problems in applied mathematics to app...

0 Michael B. Giles, et al. ∙

research

∙ 11/04/2019

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations

The main result of this article establishes strong convergence rates on ...

0 Martin Hutzenthaler, et al. ∙

research

∙ 09/30/2019

Full error analysis for the training of deep neural networks

Deep learning algorithms have been applied very successfully in recent y...

0 Christan Beck, et al. ∙

research

∙ 08/28/2019

Deep neural network approximations for Monte Carlo algorithms

Recently, it has been proposed in the literature to employ deep neural n...

0 Philipp Grohs, et al. ∙

research

∙ 08/11/2019

Space-time error estimates for deep neural network approximations for differential equations

Over the last few years deep artificial neural networks (DNNs) have very...

0 Philipp Grohs, et al. ∙

research

∙ 08/05/2019

Solving high-dimensional optimal stopping problems using deep learning

Nowadays many financial derivatives which are traded on stock and future...

0 Sebastian Becker, et al. ∙

research

∙ 07/15/2019

Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approx

One of the most challenging problems in applied mathematics is the appro...

0 Christian Beck, et al. ∙

research

∙ 07/08/2019

Deep splitting method for parabolic PDEs

In this paper we introduce a numerical method for parabolic PDEs that co...

0 Christian Beck, et al. ∙

research

∙ 05/13/2019

Towards a regularity theory for ReLU networks -- chain rule and global error estimates

Although for neural networks with locally Lipschitz continuous activatio...

0 Julius Berner, et al. ∙

research

∙ 04/02/2019

Convergence rates for the stochastic gradient descent method for non-convex objective functions

We prove the local convergence to minima and estimates on the rate of co...

0 Benjamin Fehrman, et al. ∙

research

∙ 09/09/2018

Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of B

The development of new classification and regression algorithms based on...

6 Julius Berner, et al. ∙

Arnulf Jentzen

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