Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. - Arnulf Jentzen; Peter E. Kloeden; Andreas Neuenkirch (Numerische Mathematik, 2009)

An Improved Maximum Allowable Transfer Interval for Lp -Stability of Networked Control Systems. - Arnulf Jentzen; Frank Leber; Daniela Schneisgen; Arno Berger; Stefan Siegmund (IEEE Trans. Autom. Control., 2010)

Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients. - Martin Hutzenthaler; Arnulf Jentzen (Found. Comput. Math., 2011)

Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise. - Arnulf Jentzen (SIAM J. Numer. Anal., 2011)

Galerkin Approximations for the Stochastic Burgers Equation. - Dirk Blömker; Arnulf Jentzen (SIAM J. Numer. Anal., 2013)

A Milstein Scheme for SPDEs. - Arnulf Jentzen; Michael Röckner (Found. Comput. Math., 2015)

An Exponential Wagner-Platen Type Scheme for SPDEs. - Sebastian Becker; Arnulf Jentzen; Peter E. Kloeden (SIAM J. Numer. Anal., 2016)

Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. - Weinan E; Jiequn Han; Arnulf Jentzen (CoRR, 2017)

Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning. - Jiequn Han; Arnulf Jentzen; Weinan E (CoRR, 2017)

Solving stochastic differential equations and Kolmogorov equations by means of deep learning. - Christian Beck; Sebastian Becker; Philipp Grohs; Nor Jaafari; Arnulf Jentzen (CoRR, 2018)

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. - Philipp Grohs; Fabian Hornung; Arnulf Jentzen; Philippe von Wurstemberger (CoRR, 2018)

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. - Arnulf Jentzen; Diyora Salimova; Timo Welti (CoRR, 2018)

Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. - Martin Hutzenthaler; Arnulf Jentzen; Xiaojie Wang 007 (Math. Comput., 2018)

Towards a regularity theory for ReLU networks - chain rule and global error estimates. - Julius Berner; Dennis Elbrächter; Philipp Grohs; Arnulf Jentzen (CoRR, 2019)

Solving high-dimensional optimal stopping problems using deep learning. - Sebastian Becker; Patrick Cheridito; Arnulf Jentzen; Timo Welti (CoRR, 2019)

Deep neural network approximations for Monte Carlo algorithms. - Philipp Grohs; Arnulf Jentzen; Diyora Salimova (CoRR, 2019)

Full error analysis for the training of deep neural networks. - Christian Beck; Arnulf Jentzen; Benno Kuckuck (CoRR, 2019)

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations. - Martin Hutzenthaler; Arnulf Jentzen; Felix Lindner 002; Primoz Pusnik (CoRR, 2019)

Generalised multilevel Picard approximations. - Michael B. Giles; Arnulf Jentzen; Timo Welti (CoRR, 2019)

Uniform error estimates for artificial neural network approximations for heat equations. - Lukas Gonon; Philipp Grohs; Arnulf Jentzen; David Kofler; David Siska (CoRR, 2019)

Efficient approximation of high-dimensional functions with deep neural networks. - Patrick Cheridito; Arnulf Jentzen; Florian Rossmannek (CoRR, 2019)

On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. - Mario Hefter; Arnulf Jentzen (Finance Stochastics, 2019)

Deep Optimal Stopping. - Sebastian Becker; Patrick Cheridito; Arnulf Jentzen (J. Mach. Learn. Res., 2019)

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations. - Christian Beck; Weinan E; Arnulf Jentzen (J. Nonlinear Sci., 2019)

On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations. - Weinan E; Martin Hutzenthaler; Arnulf Jentzen; Thomas Kruse (J. Sci. Comput., 2019)

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations. - Christian Beck; Lukas Gonon; Arnulf Jentzen (CoRR, 2020)

Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. - Sebastian Becker; Ramon Braunwarth; Martin Hutzenthaler; Arnulf Jentzen; Philippe von Wurstemberger (CoRR, 2020)

Space-time deep neural network approximations for high-dimensional partial differential equations. - Fabian Hornung; Arnulf Jentzen; Diyora Salimova (CoRR, 2020)

Weak error analysis for stochastic gradient descent optimization algorithms. - Aritz Bercher; Lukas Gonon; Arnulf Jentzen; Diyora Salimova (CoRR, 2020)

Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning. - Weinan E; Jiequn Han; Arnulf Jentzen (CoRR, 2020)

Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities. - Martin Hutzenthaler; Arnulf Jentzen; Thomas Kruse; Tuan Anh Nguyen (CoRR, 2020)

Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations. - Christian Beck; Arnulf Jentzen; Thomas Kruse (CoRR, 2020)

Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems. - Christian Beck; Sebastian Becker; Patrick Cheridito; Arnulf Jentzen; Ariel Neufeld (CoRR, 2020)

High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations. - Pierfrancesco Beneventano; Patrick Cheridito; Arnulf Jentzen; Philippe von Wurstemberger (CoRR, 2020)

Strong overall error analysis for the training of artificial neural networks via random initializations. - Arnulf Jentzen; Adrian Riekert (CoRR, 2020)

An overview on deep learning-based approximation methods for partial differential equations. - Christian Beck; Martin Hutzenthaler; Arnulf Jentzen; Benno Kuckuck (CoRR, 2020)

Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates. - Arnulf Jentzen; Philippe von Wurstemberger (J. Complex., 2020)

Convergence Rates for the Stochastic Gradient Descent Method for Non-Convex Objective Functions. - Benjamin J. Fehrman; Benjamin Gess; Arnulf Jentzen (J. Mach. Learn. Res., 2020)

Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. - Christian Beck; Fabian Hornung; Martin Hutzenthaler; Arnulf Jentzen; Thomas Kruse (J. Num. Math., 2020)

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions. - Arnulf Jentzen; Felix Lindner 002; Primoz Pusnik (Numer. Algorithms, 2020)

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. - Julius Berner; Philipp Grohs; Arnulf Jentzen (SIAM J. Math. Data Sci., 2020)

Convergence rates for gradient descent in the training of overparameterized artificial neural networks with biases. - Arnulf Jentzen; Timo Kröger (CoRR, 2021)

Full history recursive multilevel Picard approximations for ordinary differential equations with expectations. - Christian Beck; Martin Hutzenthaler; Arnulf Jentzen; Emilia Magnani (CoRR, 2021)

Landscape analysis for shallow ReLU neural networks: complete classification of critical points for affine target functions. - Patrick Cheridito; Arnulf Jentzen; Florian Rossmannek (CoRR, 2021)

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 (CoRR, 2021)

Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation. - Arnulf Jentzen; Adrian Riekert (CoRR, 2021)

Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation. - Simon Eberle; Arnulf Jentzen; Adrian Riekert; Georg S. Weiss (CoRR, 2021)

Strong Lp-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations. - Martin Hutzenthaler; Arnulf Jentzen; Benno Kuckuck; Joshua Lee Padgett (CoRR, 2021)

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; Adrian Riekert; Luca Scarpa (CoRR, 2021)

On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks. - Arnulf Jentzen; Adrian Riekert (CoRR, 2021)

Deep neural network approximation theory for high-dimensional functions. - Pierfrancesco Beneventano; Patrick Cheridito; Robin Graeber; Arnulf Jentzen; Benno Kuckuck (CoRR, 2021)

Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients. - Arnulf Jentzen; Ryan Kurniawan (Found. Comput. Math., 2021)

Non-convergence of stochastic gradient descent in the training of deep neural networks. - Patrick Cheridito; Arnulf Jentzen; Florian Rossmannek (J. Complex., 2021)

Solving the Kolmogorov PDE by Means of Deep Learning. - Christian Beck; Sebastian Becker; Philipp Grohs; Nor Jaafari; Arnulf Jentzen (J. Sci. Comput., 2021)

Deep Splitting Method for Parabolic PDEs. - Christian Beck; Sebastian Becker; Patrick Cheridito; Arnulf Jentzen; Ariel Neufeld (SIAM J. Sci. Comput., 2021)

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing. - Sebastian Becker; Arnulf Jentzen; Marvin S. Müller; Philippe von Wurstemberger (CoRR, 2022)

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. - Victor Boussange; Sebastian Becker; Arnulf Jentzen; Benno Kuckuck; Loïc Pellissier (CoRR, 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. - Arnulf Jentzen; Timo Kröger (CoRR, 2022)

Normalized gradient flow optimization in the training of ReLU artificial neural networks. - Simon Eberle; Arnulf Jentzen; Adrian Riekert; Georg S. Weiss (CoRR, 2022)

Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities. - Martin Hutzenthaler; Arnulf Jentzen; Thomas Kruse (Found. Comput. Math., 2022)

A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. - Patrick Cheridito; Arnulf Jentzen; Adrian Riekert; Florian Rossmannek (J. Complex., 2022)

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 (J. Mach. Learn. Res., 2022)

Landscape Analysis for Shallow Neural Networks: Complete Classification of Critical Points for Affine Target Functions. - Patrick Cheridito; Arnulf Jentzen; Florian Rossmannek (J. Nonlinear Sci., 2022)

Efficient Approximation of High-Dimensional Functions With Neural Networks. - Patrick Cheridito; Arnulf Jentzen; Florian Rossmannek (IEEE Trans. Neural Networks Learn. Syst., 2022)

Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation. - Arnulf Jentzen; Timo Welti (Appl. Math. Comput., 2023)

The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality. - Lukas Gonon; Robin Graeber; Arnulf Jentzen (CoRR, 2023)

Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations. - Arnulf Jentzen; Adrian Riekert; Philippe von Wurstemberger (CoRR, 2023)

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 Lp-sense. - Julia Ackermann; Arnulf Jentzen; Thomas Kruse; Benno Kuckuck; Joshua Lee Padgett (CoRR, 2023)

Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory. - Arnulf Jentzen; Benno Kuckuck; Philippe von Wurstemberger (CoRR, 2023)

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. - Martin Hutzenthaler; Arnulf Jentzen; Thomas Kruse; Tuan Anh Nguyen (J. Num. Math., 2023)

Space-time error estimates for deep neural network approximations for differential equations. - Philipp Grohs; Fabian Hornung; Arnulf Jentzen; Philipp Zimmermann (Adv. Comput. Math., 2023)

Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality. - Philipp Grohs; Shokhrukh Ibragimov; Arnulf Jentzen; Sarah Koppensteiner (J. Complex., 2023)

An efficient Monte Carlo scheme for Zakai equations. - Christian Beck; Sebastian Becker; Patrick Cheridito; Arnulf Jentzen; Ariel Neufeld (Commun. Nonlinear Sci. Numer. Simul., 2023)

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; Joshua Lee Padgett (CoRR, 2024)

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 (CoRR, 2024)

Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates. - Steffen Dereich; Robin Graeber; Arnulf Jentzen (CoRR, 2024)

Convergence rates for the Adam optimizer. - Steffen Dereich; Arnulf Jentzen (CoRR, 2024)

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; Siyu Liang; Adrian Riekert; Philippe von Wurstemberger (CoRR, 2024)

Non-convergence to global minimizers in data driven supervised deep learning: Adam and stochastic gradient descent optimization provably fail to converge to global minimizers in the training of deep neural networks with ReLU activation. - Sonja Hannibal; Arnulf Jentzen; Do Minh Thang (CoRR, 2024)

An overview of diffusion models for generative artificial intelligence. - Davide Gallon; Arnulf Jentzen; Philippe von Wurstemberger (CoRR, 2024)

On the Existence of Minimizers in Shallow Residual ReLU Neural Network Optimization Landscapes. - Steffen Dereich; Arnulf Jentzen; Sebastian Kassing (SIAM J. Numer. Anal., 2024)

Gradient Descent Provably Escapes Saddle Points in the Training of Shallow ReLU Networks. - Patrick Cheridito; Arnulf Jentzen; Florian Rossmannek (J. Optim. Theory Appl., 2024)

Weak convergence rates for temporal numerical approximations of the semilinear stochastic wave equation with multiplicative noise. - Sonja Cox; Arnulf Jentzen; Felix Lindner 002 (Numerische Mathematik, 2024)

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 (SIAM/ASA J. Uncertain. Quantification, 2025)

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 (CoRR, 2025)

Mathematical analysis of the gradients in deep learning. - Steffen Dereich; Thang Do; Arnulf Jentzen; Frederic Weber (CoRR, 2025)

On the logical skills of large language models: evaluations using arbitrarily complex first-order logic problems. - Shokhrukh Ibragimov; Arnulf Jentzen; Benno Kuckuck (CoRR, 2025)

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 (CoRR, 2025)

Sharp higher order convergence rates for the Adam optimizer. - Steffen Dereich; Arnulf Jentzen; Adrian Riekert (CoRR, 2025)

SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures. - Julian Kranz; Davide Gallon; Steffen Dereich; Arnulf Jentzen (CoRR, 2025)

A brief review of the Deep BSDE method for solving high-dimensional partial differential equations. - Jiequn Han; Arnulf Jentzen; Weinan E (CoRR, 2025)

PADAM: Parallel averaged Adam reduces the error for stochastic optimization in scientific machine learning. - Arnulf Jentzen; Julian Kranz; Adrian Riekert (CoRR, 2025)

Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality. - Arnulf Jentzen; Konrad Kleinberg; Thomas Kruse (CoRR, 2025)