Three ways to solve partial differential equations with neural networks—A review
J Blechschmidt, OG Ernst - GAMM‐Mitteilungen, 2021 - Wiley Online Library
Neural networks are increasingly used to construct numerical solution methods for partial
differential equations. In this expository review, we introduce and contrast three important …
differential equations. In this expository review, we introduce and contrast three important …
An overview on deep learning-based approximation methods for partial differential equations
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 …
Learning the solution operator of parametric partial differential equations with physics-informed DeepONets
Partial differential equations (PDEs) play a central role in the mathematical analysis and
modeling of complex dynamic processes across all corners of science and engineering …
modeling of complex dynamic processes across all corners of science and engineering …
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 …
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning
In recent years, tremendous progress has been made on numerical algorithms for solving
partial differential equations (PDEs) in a very high dimension, using ideas from either …
partial differential equations (PDEs) in a very high dimension, using ideas from either …
Long-time integration of parametric evolution equations with physics-informed deeponets
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing
and simulating complex dynamic processes across all corners of science and engineering …
and simulating complex dynamic processes across all corners of science and engineering …
[BOOK][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 …
A theoretical analysis of deep neural networks and parametric PDEs
We derive upper bounds on the complexity of ReLU neural networks approximating the
solution maps of parametric partial differential equations. In particular, without any …
solution maps of parametric partial differential equations. In particular, without any …
Error bounds for approximations with deep ReLU neural networks in norms
We analyze to what extent deep Rectified Linear Unit (ReLU) neural networks can efficiently
approximate Sobolev regular functions if the approximation error is measured with respect to …
approximate Sobolev regular functions if the approximation error is measured with respect to …
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
Deep neural networks and other deep learning methods have very successfully been
applied to the numerical approximation of high-dimensional nonlinear parabolic partial …
applied to the numerical approximation of high-dimensional nonlinear parabolic partial …