Neural operators for accelerating scientific simulations and design
Scientific discovery and engineering design are currently limited by the time and cost of
physical experiments. Numerical simulations are an alternative approach but are usually …
physical experiments. Numerical simulations are an alternative approach but are usually …
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 …
基于神经网络的偏微分方程求解方法研究综述
查文舒， 李道伦， 沈路航， 张雯， 刘旭亮 - 力学学报, 2022 - lxxb.cstam.org.cn
神经网络作为一种强大的信息处理工具在计算机视觉, 生物医学, 油气工程领域得到广泛应用,
引发多领域技术变革. 深度学习网络具有非常强的学习能力, 不仅能发现物理规律 …
引发多领域技术变革. 深度学习网络具有非常强的学习能力, 不仅能发现物理规律 …
Adaptive deep neural networks methods for high-dimensional partial differential equations
S Zeng, Z Zhang, Q Zou - Journal of Computational Physics, 2022 - Elsevier
We present three adaptive techniques to improve the computational performance of deep
neural network (DNN) methods for high-dimensional partial differential equations (PDEs) …
neural network (DNN) methods for high-dimensional partial differential equations (PDEs) …
Approximation rates for neural networks with encodable weights in smoothness spaces
We examine the necessary and sufficient complexity of neural networks to approximate
functions from different smoothness spaces under the restriction of encodable network …
functions from different smoothness spaces under the restriction of encodable network …
[HTML][HTML] Numerical solution of the parametric diffusion equation by deep neural networks
M Geist, P Petersen, M Raslan, R Schneider… - Journal of Scientific …, 2021 - Springer
We perform a comprehensive numerical study of the effect of approximation-theoretical
results for neural networks on practical learning problems in the context of numerical …
results for neural networks on practical learning problems in the context of numerical …
An extreme learning machine-based method for computational PDEs in higher dimensions
We present two effective methods for solving high-dimensional partial differential equations
(PDE) based on randomized neural networks. Motivated by the universal approximation …
(PDE) based on randomized neural networks. Motivated by the universal approximation …
Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients
We address a new numerical method based on machine learning and in particular based on
the concept of the so-called Extreme Learning Machines, to approximate the solution of …
the concept of the so-called Extreme Learning Machines, to approximate the solution of …
Actor-critic method for high dimensional static Hamilton--Jacobi--Bellman partial differential equations based on neural networks
We propose a novel numerical method for high dimensional Hamilton--Jacobi--Bellman
(HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as …
(HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as …
[HTML][HTML] Data-driven modeling of geometry-adaptive steady heat conduction based on convolutional neural networks
A data-driven model for rapid prediction of the steady-state heat conduction of a hot object
with arbitrary geometry is developed. Mathematically, the steady-state heat conduction can …
with arbitrary geometry is developed. Mathematically, the steady-state heat conduction can …