Arnulf Jentzen

Publication Activity (10 Years)

Years Active: 2009-2024
Publications (10 Years): 89

Top Topics

Top Venues

Found. Comput. Math.

J. Mach. Learn. Res.

Publications

Arnulf Jentzen, Adrian Riekert
Non-convergence to global minimizers for Adam and stochastic gradient descent optimization and constructions of local minimizers in the training of artificial neural networks. CoRR (2024)
Julia Ackermann, Arnulf Jentzen, Benno Kuckuck, Joshua Lee Padgett
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. CoRR (2024)
Steffen Dereich, Robin Graeber, Arnulf Jentzen
Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates. CoRR (2024)
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses. CoRR (2024)
Steffen Dereich, Arnulf Jentzen
Convergence rates for the Adam optimizer. CoRR (2024)
Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld
An efficient Monte Carlo scheme for Zakai equations. Commun. Nonlinear Sci. Numer. Simul. 126 (2023)
Arnulf Jentzen, Timo Welti
Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation. Appl. Math. Comput. 455 (2023)
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. J. Num. Math. 31 (1) (2023)
Lukas Gonon, Robin Graeber, Arnulf Jentzen
The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality. CoRR (2023)
Julia Ackermann, Arnulf Jentzen, Thomas Kruse, Benno Kuckuck, Joshua Lee Padgett
-sense. CoRR (2023)
Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philipp Zimmermann
Space-time error estimates for deep neural network approximations for differential equations. Adv. Comput. Math. 49 (1) (2023)
Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger
Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory. CoRR (2023)
Steffen Dereich, Arnulf Jentzen, Sebastian Kassing
On the existence of minimizers in shallow residual ReLU neural network optimization landscapes. CoRR (2023)
Arnulf Jentzen, Adrian Riekert, Philippe von Wurstemberger
Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations. CoRR (2023)
Philipp Grohs, Shokhrukh Ibragimov, Arnulf Jentzen, Sarah Koppensteiner
Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality. J. Complex. 77 (2023)
Simon Eberle, Arnulf Jentzen, Adrian Riekert, Georg S. Weiss
Normalized gradient flow optimization in the training of ReLU artificial neural networks. CoRR (2022)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Gradient descent provably escapes saddle points in the training of shallow ReLU networks. CoRR (2022)
Sebastian Becker, Arnulf Jentzen, Marvin S. Müller, Philippe von Wurstemberger
Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing. CoRR (2022)
Patrick Cheridito, Arnulf Jentzen, Adrian Riekert, Florian Rossmannek
A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. J. Complex. 72 (2022)
Victor Boussange, Sebastian Becker, Arnulf Jentzen, Benno Kuckuck, Loïc Pellissier
Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. CoRR (2022)
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities. Found. Comput. Math. 22 (4) (2022)
Arnulf Jentzen, Adrian Riekert
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. J. Mach. Learn. Res. 23 (2022)
Arnulf Jentzen, Timo Kröger
On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector. CoRR (2022)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Landscape Analysis for Shallow Neural Networks: Complete Classification of Critical Points for Affine Target Functions. J. Nonlinear Sci. 32 (5) (2022)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Efficient Approximation of High-Dimensional Functions With Neural Networks. IEEE Trans. Neural Networks Learn. Syst. 33 (7) (2022)
Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld
An efficient Monte Carlo scheme for Zakai equations. CoRR (2022)
Pierfrancesco Beneventano, Patrick Cheridito, Robin Graeber, Arnulf Jentzen, Benno Kuckuck
Deep neural network approximation theory for high-dimensional functions. CoRR (2021)
Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck, Joshua Lee Padgett
-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations. CoRR (2021)
Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld
Deep Splitting Method for Parabolic PDEs. SIAM J. Sci. Comput. 43 (5) (2021)
Arnulf Jentzen, Ryan Kurniawan
Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients. Found. Comput. Math. 21 (2) (2021)
Christian Beck, Sebastian Becker, Philipp Grohs, Nor Jaafari, Arnulf Jentzen
Solving the Kolmogorov PDE by Means of Deep Learning. J. Sci. Comput. 88 (3) (2021)
Arnulf Jentzen, Adrian Riekert
A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions. CoRR (2021)
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. CoRR (2021)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Non-convergence of stochastic gradient descent in the training of deep neural networks. J. Complex. 64 (2021)
Arnulf Jentzen, Adrian Riekert
On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks. CoRR (2021)
Philipp Grohs, Shokhrukh Ibragimov, Arnulf Jentzen, Sarah Koppensteiner
Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality. CoRR (2021)
Patrick Cheridito, Arnulf Jentzen, Adrian Riekert, Florian Rossmannek
A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. CoRR (2021)
Simon Eberle, Arnulf Jentzen, Adrian Riekert, Georg S. Weiss
Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation. CoRR (2021)
Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Emilia Magnani
Full history recursive multilevel Picard approximations for ordinary differential equations with expectations. CoRR (2021)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Landscape analysis for shallow ReLU neural networks: complete classification of critical points for affine target functions. CoRR (2021)
Arnulf Jentzen, Adrian Riekert
Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation. CoRR (2021)
Martin Hutzenthaler, Arnulf Jentzen, Katharina Pohl, Adrian Riekert, Luca Scarpa
Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions. CoRR (2021)
Arnulf Jentzen, Adrian Riekert
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. CoRR (2021)
Arnulf Jentzen, Timo Kröger
Convergence rates for gradient descent in the training of overparameterized artificial neural networks with biases. CoRR (2021)
Fabian Hornung, Arnulf Jentzen, Diyora Salimova
Space-time deep neural network approximations for high-dimensional partial differential equations. CoRR (2020)
Christian Beck, Lukas Gonon, Arnulf Jentzen
Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations. CoRR (2020)
Arnulf Jentzen, Philippe von Wurstemberger
Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates. J. Complex. 57 (2020)
Christian Beck, Arnulf Jentzen, Thomas Kruse
Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations. CoRR (2020)
Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck
An overview on deep learning-based approximation methods for partial differential equations. CoRR (2020)
Julius Berner, Philipp Grohs, Arnulf Jentzen
Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations. SIAM J. Math. Data Sci. 2 (3) (2020)
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities. CoRR (2020)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Non-convergence of stochastic gradient descent in the training of deep neural networks. CoRR (2020)
Benjamin J. Fehrman, Benjamin Gess, Arnulf Jentzen
Convergence Rates for the Stochastic Gradient Descent Method for Non-Convex Objective Functions. J. Mach. Learn. Res. 21 (2020)
Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld
Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems. CoRR (2020)
Aritz Bercher, Lukas Gonon, Arnulf Jentzen, Diyora Salimova
Weak error analysis for stochastic gradient descent optimization algorithms. CoRR (2020)
Weinan E, Jiequn Han, Arnulf Jentzen
Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning. CoRR (2020)
Arnulf Jentzen, Adrian Riekert
Strong overall error analysis for the training of artificial neural networks via random initializations. CoRR (2020)
Pierfrancesco Beneventano, Patrick Cheridito, Arnulf Jentzen, Philippe von Wurstemberger
High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations. CoRR (2020)
Arnulf Jentzen, Timo Welti
Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation. CoRR (2020)
Sebastian Becker, Ramon Braunwarth, Martin Hutzenthaler, Arnulf Jentzen, Philippe von Wurstemberger
Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. CoRR (2020)
Christian Beck, Fabian Hornung, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. J. Num. Math. 28 (4) (2020)
Arnulf Jentzen, Felix Lindner, Primoz Pusnik
Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions. Numer. Algorithms 85 (4) (2020)
Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philipp Zimmermann
Space-time error estimates for deep neural network approximations for differential equations. CoRR (2019)
Christian Beck, Fabian Hornung, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. CoRR (2019)
Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Timo Welti
Solving high-dimensional optimal stopping problems using deep learning. CoRR (2019)
Sebastian Becker, Patrick Cheridito, Arnulf Jentzen
Deep Optimal Stopping. J. Mach. Learn. Res. 20 (2019)
Martin Hutzenthaler, Arnulf Jentzen, Felix Lindner, Primoz Pusnik
Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations. CoRR (2019)
Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations. J. Sci. Comput. 79 (3) (2019)
Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek
Efficient approximation of high-dimensional functions with deep neural networks. CoRR (2019)
Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld
Deep splitting method for parabolic PDEs. CoRR (2019)
Michael B. Giles, Arnulf Jentzen, Timo Welti
Generalised multilevel Picard approximations. CoRR (2019)
Christian Beck, Arnulf Jentzen, Benno Kuckuck
Full error analysis for the training of deep neural networks. CoRR (2019)
Christian Beck, Weinan E, Arnulf Jentzen
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations. J. Nonlinear Sci. 29 (4) (2019)
Benjamin J. Fehrman, Benjamin Gess, Arnulf Jentzen
Convergence rates for the stochastic gradient descent method for non-convex objective functions. CoRR (2019)
Mario Hefter, Arnulf Jentzen
On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. Finance Stochastics 23 (1) (2019)
Lukas Gonon, Philipp Grohs, Arnulf Jentzen, David Kofler, David Siska
Uniform error estimates for artificial neural network approximations for heat equations. CoRR (2019)
Julius Berner, Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen
Towards a regularity theory for ReLU networks - chain rule and global error estimates. CoRR (2019)
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. CoRR (2019)
Philipp Grohs, Arnulf Jentzen, Diyora Salimova
Deep neural network approximations for Monte Carlo algorithms. CoRR (2019)
Martin Hutzenthaler, Arnulf Jentzen, Xiaojie Wang
Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math. Comput. 87 (311) (2018)
Julius Berner, Philipp Grohs, Arnulf Jentzen
Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. CoRR (2018)
Christian Beck, Sebastian Becker, Philipp Grohs, Nor Jaafari, Arnulf Jentzen
Solving stochastic differential equations and Kolmogorov equations by means of deep learning. CoRR (2018)
Arnulf Jentzen, Diyora Salimova, Timo Welti
A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients. CoRR (2018)
Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger
A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. CoRR (2018)
Jiequn Han, Arnulf Jentzen, Weinan E
Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning. CoRR (2017)
Christian Beck, Weinan E, Arnulf Jentzen
Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. CoRR (2017)
Weinan E, Jiequn Han, Arnulf Jentzen
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. CoRR (2017)
Sebastian Becker, Arnulf Jentzen, Peter E. Kloeden
An Exponential Wagner-Platen Type Scheme for SPDEs. SIAM J. Numer. Anal. 54 (4) (2016)
Arnulf Jentzen, Michael Röckner
A Milstein Scheme for SPDEs. Found. Comput. Math. 15 (2) (2015)
Dirk Blömker, Arnulf Jentzen
Galerkin Approximations for the Stochastic Burgers Equation. SIAM J. Numer. Anal. 51 (1) (2013)
Arnulf Jentzen
Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise. SIAM J. Numer. Anal. 49 (2) (2011)
Martin Hutzenthaler, Arnulf Jentzen
Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients. Found. Comput. Math. 11 (6) (2011)
Arnulf Jentzen, Frank Leber, Daniela Schneisgen, Arno Berger, Stefan Siegmund
-Stability of Networked Control Systems. IEEE Trans. Autom. Control. 55 (1) (2010)
Arnulf Jentzen, Peter E. Kloeden, Andreas Neuenkirch
Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. Numerische Mathematik 112 (1) (2009)