An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations

被引：0

作者：

Mishra, Abhishek ^{[1
]}

Anitescu, Cosmin ^{[2
]}

Budarapu, Pattabhi Ramaiah ^{[1
]}

Natarajan, Sundararajan ^{[3
]}

Vundavilli, Pandu Ranga ^{[1
]}

Rabczuk, Timon ^{[2
]}

机构：

[1] Indian Inst Technol, Sch Mech Sci, Bhubaneswar 752050, India

[2] Bauhaus Univ Weimar, Inst Struct Mech, D-99423 Weimar, Germany

[3] Indian Inst Technol Madras, Dept Mech Engn, Chennai 600036, India

来源：

FRONTIERS OF STRUCTURAL AND CIVIL ENGINEERING | 2024年 / 18卷 / 08期

关键词：

collocation method; artificial neural networks; deep machine learning; Sine-Gordon equation; transient wave equation; dynamic scalar and elasto-dynamic equation; Runge-Kutta method; ALGORITHM;

D O I：

10.1007/s11709-024-1011-4

中图分类号：

TU [建筑科学];

学科分类号：

0813 ;

摘要：

A combined deep machine learning (DML) and collocation based approach to solve the partial differential equations using artificial neural networks is proposed. The developed method is applied to solve problems governed by the Sine-Gordon equation (SGE), the scalar wave equation and elasto-dynamics. Two methods are studied: one is a space-time formulation and the other is a semi-discrete method based on an implicit Runge-Kutta (RK) time integration. The methodology is implemented using the Tensorflow framework and it is tested on several numerical examples. Based on the results, the relative normalized error was observed to be less than 5% in all cases.

引用

页码：1296 / 1310

页数：15

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