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

9 September 2018

Papers citing "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"

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Title
Super-fast rates of convergence for Neural Networks Classifiers under the Hard Margin Condition Nathanael Tepakbong Ding-Xuan Zhou Xiang Zhou 39 0 0 13 May 2025
Solving Poisson Equations using Neural Walk-on-Spheres Hong Chul Nam Julius Berner Anima Anandkumar 37 3 0 05 Jun 2024
$A numerical approach for the fractional Laplacian via deep neural networks$ A numerical approach for the fractional Laplacian via deep neural networks Nicolás Valenzuela 34 3 0 30 Aug 2023
Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time Y. Qian Yongchao Zhang Yuanfei Huang S. Dong PINN 21 1 0 22 Mar 2023
Deep neural network expressivity for optimal stopping problems Lukas Gonon 27 6 0 19 Oct 2022
Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces Johannes Milz T. Surowiec 26 4 0 29 Jul 2022
Error analysis for deep neural network approximations of parametric hyperbolic conservation laws Tim De Ryck Siddhartha Mishra PINN 15 10 0 15 Jul 2022
Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning Lorenz Richter Julius Berner 27 19 0 21 Jun 2022
Learning ReLU networks to high uniform accuracy is intractable Julius Berner Philipp Grohs F. Voigtlaender 32 4 0 26 May 2022
Convergence of a robust deep FBSDE method for stochastic control Kristoffer Andersson Adam Andersson C. Oosterlee 34 19 0 18 Jan 2022
Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces Hao Liu Haizhao Yang Minshuo Chen T. Zhao Wenjing Liao 32 36 0 01 Jan 2022
Optimal learning of high-dimensional classification problems using deep neural networks P. Petersen F. Voigtlaender 33 10 0 23 Dec 2021
Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems Nikolas Nusken Lorenz Richter PINN DiffM 31 27 0 07 Dec 2021
Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion S. Basir Inanc Senocak PINN AI4CE 34 68 0 30 Sep 2021
Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs Tim De Ryck Siddhartha Mishra PINN 21 100 0 28 Jun 2021
Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality Lukas Gonon 21 35 0 14 Jun 2021
Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems Clemens Hutter R. Gül Helmut Bölcskei 21 9 0 06 May 2021
Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces Philipp Grohs F. Voigtlaender 16 34 0 06 Apr 2021
Quantitative approximation results for complex-valued neural networks A. Caragea D. Lee J. Maly G. Pfander F. Voigtlaender 13 5 0 25 Feb 2021
Reproducing Activation Function for Deep Learning Senwei Liang Liyao Lyu Chunmei Wang Haizhao Yang 36 21 0 13 Jan 2021
A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations Jianfeng Lu Yulong Lu Min Wang 36 37 0 05 Jan 2021
An overview on deep learning-based approximation methods for partial differential equations C. Beck Martin Hutzenthaler Arnulf Jentzen Benno Kuckuck 30 146 0 22 Dec 2020
Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data Ben Adcock Simone Brugiapaglia N. Dexter S. Moraga 36 29 0 11 Dec 2020
Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes Raffaele Marino N. Macris 19 16 0 09 Dec 2020
Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions Philipp Grohs L. Herrmann 30 52 0 10 Jul 2020
Expressivity of Deep Neural Networks Ingo Gühring Mones Raslan Gitta Kutyniok 16 51 0 09 Jul 2020
Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory Tao Luo Haizhao Yang 32 73 0 28 Jun 2020
Space-time deep neural network approximations for high-dimensional partial differential equations F. Hornung Arnulf Jentzen Diyora Salimova AI4CE 24 19 0 03 Jun 2020
ReLU Code Space: A Basis for Rating Network Quality Besides Accuracy Natalia Shepeleva Werner Zellinger Michal Lewandowski Bernhard A. Moser 22 3 0 20 May 2020
Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space Nikolas Nusken Lorenz Richter AI4CE 14 103 0 11 May 2020
Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation Arnulf Jentzen Timo Welti 17 15 0 03 Mar 2020
Uniform error estimates for artificial neural network approximations for heat equations Lukas Gonon Philipp Grohs Arnulf Jentzen David Kofler David Siska 29 34 0 20 Nov 2019
A deep surrogate approach to efficient Bayesian inversion in PDE and integral equation models Teo Deveney Amelia Gosse Peter Du 23 9 0 03 Oct 2019
Space-time error estimates for deep neural network approximations for differential equations Philipp Grohs F. Hornung Arnulf Jentzen Philipp Zimmermann 26 33 0 11 Aug 2019
Deep splitting method for parabolic PDEs C. Beck S. Becker Patrick Cheridito Arnulf Jentzen Ariel Neufeld 23 125 0 08 Jul 2019
A Theoretical Analysis of Deep Neural Networks and Parametric PDEs Gitta Kutyniok P. Petersen Mones Raslan R. Schneider 20 197 0 31 Mar 2019
Deep Neural Network Approximation Theory Dennis Elbrächter Dmytro Perekrestenko Philipp Grohs Helmut Bölcskei 14 207 0 08 Jan 2019
Unbiased deep solvers for linear parametric PDEs Marc Sabate Vidales David Siska Lukasz Szpruch OOD 26 7 0 11 Oct 2018
Solving the Kolmogorov PDE by means of deep learning C. Beck S. Becker Philipp Grohs Nor Jaafari Arnulf Jentzen 11 91 0 01 Jun 2018
Optimal Approximation with Sparsely Connected Deep Neural Networks Helmut Bölcskei Philipp Grohs Gitta Kutyniok P. Petersen 27 255 0 04 May 2017
The Loss Surfaces of Multilayer Networks A. Choromańska Mikael Henaff Michaël Mathieu Gerard Ben Arous Yann LeCun ODL 183 1,185 0 30 Nov 2014
Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes Ohad Shamir Tong Zhang 101 571 0 08 Dec 2012