A Local Deep Learning Method for Solving High Order Partial Differential Equations

Yang，Jiang1,2,3; Zhu，Quanhui2

2022

10.4208/NMTMA.OA-2021-0035

Numerical Mathematics-Theory Methods and Applications   Impact Factor & Quartile

1004-8979

2079-7338

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly, and high order derivatives lack robustness for training purposes. We propose a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables to rewrite the PDEs into a system of low order differential equations as what is done in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural network. By taking the residual of the system as a loss function, we can optimize the network parameters to approximate the solution. The whole process relies on low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.

Deep Galerkin method Deep learning Deep neural network High order PDEs Reduction of order

SCI

English

First ; Corresponding

National Natural Science Foundation of China-Guangdong Joint Fund[11961160718];Applied Basic Research Foundation of Yunnan Province[2018A0303130123];Guangdong Provincial Key Laboratory of Urology[2019B030301001];National Natural Science Foundation of China[NSFC-11871264];

WOS:000723826100001

2-s2.0-85125719766

Scopus

1.International Center of Mathematics,Southern University of Science and Technology,Shenzhen,518055,China
2.Department of Mathematics,Southern University of Science and Technology,Shenzhen,518055,China
3.Guangdong Provincial Key Laboratory of Computational Science and Material Design,Southern University of Science and Technology,Shenzhen,518055,China

Yang，Jiang,Zhu，Quanhui. A Local Deep Learning Method for Solving High Order Partial Differential Equations[J]. Numerical Mathematics-Theory Methods and Applications,2022,15(1):42-67.

Yang，Jiang,&Zhu，Quanhui.(2022).A Local Deep Learning Method for Solving High Order Partial Differential Equations.Numerical Mathematics-Theory Methods and Applications,15(1),42-67.

Yang，Jiang,et al."A Local Deep Learning Method for Solving High Order Partial Differential Equations".Numerical Mathematics-Theory Methods and Applications 15.1(2022):42-67.