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
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 …
[HTML][HTML] Tackling the curse of dimensionality with physics-informed neural networks
The curse-of-dimensionality taxes computational resources heavily with exponentially
increasing computational cost as the dimension increases. This poses great challenges in …
increasing computational cost as the dimension increases. This poses great challenges in …
Generic bounds on the approximation error for physics-informed (and) operator learning
We propose a very general framework for deriving rigorous bounds on the approximation
error for physics-informed neural networks (PINNs) and operator learning architectures such …
error for physics-informed neural networks (PINNs) and operator learning architectures such …
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 …
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 …
[HTML][HTML] Hutchinson trace estimation for high-dimensional and high-order physics-informed neural networks
Abstract Physics-Informed Neural Networks (PINNs) have proven effective in solving partial
differential equations (PDEs), especially when some data are available by seamlessly …
differential equations (PDEs), especially when some data are available by seamlessly …
Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
Optimal control of diffusion processes is intimately connected to the problem of solving
certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired …
certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired …
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 …
Bias-variance trade-off in physics-informed neural networks with randomized smoothing for high-dimensional PDEs
While physics-informed neural networks (PINNs) have been proven effective for low-
dimensional partial differential equations (PDEs), the computational cost remains a hurdle in …
dimensional partial differential equations (PDEs), the computational cost remains a hurdle in …