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 …
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 …
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 …
[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 …
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 …
Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard …
C Beck, F Hornung, M Hutzenthaler… - Journal of Numerical …, 2020 - degruyter.com
One of the most challenging problems in applied mathematics is the approximate solution of
nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic …
nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic …
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 …
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with …
Recently, several deep learning (DL) methods for approximating high-dimensional partial
differential equations (PDEs) have been proposed. The interest that these methods have …
differential equations (PDEs) have been proposed. The interest that these methods have …
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real-
world phenomena. In a number of such real-world phenomena the PDEs under …
world phenomena. In a number of such real-world phenomena the PDEs under …