Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms

DOI10.1007/s42985-021-00100-zzbMath1490.65232arXiv1908.10828OpenAlexW4281674439WikidataQ114216504 ScholiaQ114216504MaRDI QIDQ2152480

Philipp Grohs, Diyora Salimova, Arnulf Jentzen

Publication date: 8 July 2022

Published in: SN Partial Differential Equations and Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1908.10828

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Monte Carlo methods (65C05) Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65M99)

Related Items (11)

A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations ⋮ Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities ⋮ Unbiased Deep Solvers for Linear Parametric PDEs ⋮ Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs ⋮ Deep learning methods for partial differential equations and related parameter identification problems ⋮ Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation ⋮ Lower bounds for artificial neural network approximations: a proof that shallow neural networks fail to overcome the curse of dimensionality ⋮ Deep Weak Approximation of SDEs: A Spatial Approximation Scheme for Solving Kolmogorov Equations ⋮ Mobility Estimation for Langevin Dynamics Using Control Variates ⋮ Approximation properties of residual neural networks for Kolmogorov PDEs ⋮ An overview on deep learning-based approximation methods for partial differential equations

Uses Software

• ImageNet
• DeepFace
• PDE-Net
• AlexNet
• DGM

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