It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning …
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) …
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network …
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 …
Y Wang, S Dong - Computer Methods in Applied Mechanics and …, 2024 - Elsevier
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation …
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 …
M Zhou, J Han, J Lu - SIAM Journal on Scientific Computing, 2021 - SIAM
We propose a novel numerical method for high dimensional Hamilton--Jacobi--Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as …
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 …