A deep learning method for solving high-order nonlinear soliton equations

被引：12

作者：

Cui, Shikun ^{[1
]}

Wang, Zhen ^{[1
,2
,3
]}

Han, Jiaqi ^{[1
]}

Cui, Xinyu ^{[1
]}

Meng, Qicheng ^{[2
]}

机构：

[1] Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China

[2] Minist Nat Resources, Inst Oceanog 2, State Key Lab Satellite Ocean Environm Dynam, Hangzhou 310000, Peoples R China

[3] Key Lab Computat Math & Data Intelligence Liaonin, Dalian 116024, Peoples R China

来源：

COMMUNICATIONS IN THEORETICAL PHYSICS | 2022年 / 74卷 / 07期

基金：

美国国家科学基金会;

关键词：

deep learning method; physics-informed neural networks; high-order nonlinear soliton equations; interaction between solitons; the numerical driven solution; GENERALIZED BOUSSINESQ EQUATION; HIROTA 3-SOLITON CONDITION; NUMERICAL-SOLUTION; WAVE SOLUTIONS; SEARCH;

D O I：

10.1088/1572-9494/ac7202

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higher-order nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg-de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.

引用

页数：13

共 35 条

[1] An efficient approach for the numerical solution of fifth-order KdV equations [J].

Ahmad, Hijaz ;

Khan, Tufail A. ;

Yao, Shao-Wen .

OPEN MATHEMATICS, 2020, 18 :738-748

[2] RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION [J].

Clarkson, Peter A. .

ANALYSIS AND APPLICATIONS, 2008, 6 (04) :349-369

[3]

Guo BX, 2016, COMMUN THEOR PHYS, V66, P589, DOI 10.1088/0253-6102/66/6/589

[4] Solving high-dimensional partial differential equations using deep learning [J].

Han, Jiequn ;

Jentzen, Arnulf ;

Weinan, E. .

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 2018, 115 (34) :8505-8510

[5] A SEARCH FOR BILINEAR EQUATIONS PASSING HIROTA 3-SOLITON CONDITION .1. KDV-TYPE BILINEAR EQUATIONS [J].

HIETARINTA, J .

JOURNAL OF MATHEMATICAL PHYSICS, 1987, 28 (08) :1732-1742

[6] A SEARCH FOR BILINEAR EQUATIONS PASSING HIROTA 3-SOLITON CONDITION .2. MKDV-TYPE BILINEAR EQUATIONS [J].

HIETARINTA, J .

JOURNAL OF MATHEMATICAL PHYSICS, 1987, 28 (09) :2094-2101

[7] On the Initial-Boundary Value Problem for the Linearized Boussinesq Equation [J].

Himonas, A. Alexandrou ;

Mantzavinos, Dionyssios .

STUDIES IN APPLIED MATHEMATICS, 2015, 134 (01) :62-100

[8] Multi-symplectic method for generalized fifth-order KdV equation [J].

Hu Wei-Peng ;

Deng Zi-Chen .

CHINESE PHYSICS B, 2008, 17 (11) :3923-3929

[9] An explicit and numerical solutions of some fifth-order KdV equation by decomposition method [J].

Kaya, D .

APPLIED MATHEMATICS AND COMPUTATION, 2003, 144 (2-3) :353-363

[10] NUMERICAL-SOLUTION OF KDV EQUATION [J].

KUO, PY ;

WU, HM .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1981, 82 (02) :334-345

← 1 2 3 4 →