An overview on deep learning-based approximation methods for partial differential equations
C Beck, M Hutzenthaler, A Jentzen… - arXiv preprint arXiv …, 2020 - arxiv.org
It is one of the most challenging problems in applied mathematics to approximatively solve
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
The modern mathematics of deep learning
We describe the new field of the mathematical analysis of deep learning. This field emerged
around a list of research questions that were not answered within the classical framework of …
around a list of research questions that were not answered within the classical framework of …
[图书][B] A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations
Artificial neural networks (ANNs) have very successfully been used in numerical simulations
for a series of computational problems ranging from image classification/image recognition …
for a series of computational problems ranging from image classification/image recognition …
Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical …
The development of new classification and regression algorithms based on empirical risk
minimization (ERM) over deep neural network hypothesis classes, coined deep learning …
minimization (ERM) over deep neural network hypothesis classes, coined deep learning …
Solving the Kolmogorov PDE by means of deep learning
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations
(PDEs) associated to them have been widely used in models from engineering, finance, and …
(PDEs) associated to them have been widely used in models from engineering, finance, and …
Deep splitting method for parabolic PDEs
C Beck, S Becker, P Cheridito, A Jentzen… - SIAM Journal on Scientific …, 2021 - SIAM
In this paper, we introduce a numerical method for nonlinear parabolic partial differential
equations (PDEs) that combines operator splitting with deep learning. It divides the PDE …
equations (PDEs) that combines operator splitting with deep learning. It divides the PDE …
A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with …
In recent years deep artificial neural networks (DNNs) have been successfully employed in
numerical simulations for a multitude of computational problems including, for example …
numerical simulations for a multitude of computational problems including, for example …
[图书][B] Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients
M Hutzenthaler, A Jentzen - 2015 - ams.org
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing
nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the …
nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the …
DNN expression rate analysis of high-dimensional PDEs: application to option pricing
We analyze approximation rates by deep ReLU networks of a class of multivariate solutions
of Kolmogorov equations which arise in option pricing. Key technical devices are deep …
of Kolmogorov equations which arise in option pricing. Key technical devices are deep …
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
For a long time it has been well-known that high-dimensional linear parabolic partial
differential equations (PDEs) can be approximated by Monte Carlo methods with a …
differential equations (PDEs) can be approximated by Monte Carlo methods with a …