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International Journal of Computational MethodsOnline Ready
No Access


A PHYSICS-INFORMED RECURRENT NEURAL NETWORK FOR SOLVING TIME-DEPENDENT PARTIAL
DIFFERENTIAL EQUATIONS

 * Ying Liang, 
 * Ruiping Niu, 
 * Junhong Yue, and 
 * Min lei

Ying Liang

https://orcid.org/0000-0002-5476-8846

College of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi
030024, P. R. China



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, 
Ruiping Niu

College of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi
030024, P. R. China



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, 
Junhong Yue

College of Big Data, Taiyuan University of Technology, Taiyuan, Shanxi 030024,
P. R. China



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, and 
Min lei

https://orcid.org/0000-0002-1900-3223

College of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi
030024, P. R. China

E-mail Address: leimin@tyut.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0219876223410037Cited by:0

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ABSTRACT

In this paper, a physics-informed recurrent neural network (PIRNN) is proposed
to solve time-dependent partial differential equations (PDEs), which devices
LSTM cells to ensure the continuity of field variables in time stepping. The
number of the training parameters is sharply reduced due to the parameter
sharing implemented in LSTM cells so that the efficiency of PIRNN greatly
improves. In order to preferably simulate the physical process and improve the
accuracy of prediction, the predicted values of the current layer are employed
as the input of the next layer, which exploits the idea of FDM. Thus, more
information can be applied for the next prediction, and the field values at
different time steps can be obtained as well. Besides, the loss value of the
governing equation, the loss value of the initial condition and the loss value
of the boundary condition are used to construct the loss function so that the
physical law is also fully utilized. Finally, we conduct the heat conduction
equation, wave equation and 2D Burgers equation to demonstrate the performance
of PIRNN. Numerical experiments show that the proposed PIRNN can accurately and
efficiently predict the field values at any time, in which nonuniform time steps
can be used and the error accumulation is avoided.


Keywords:
 * Time-dependent PDE
 * recurrent neural network
 * long short-term memory
 * physics-informed neural network
 * finite difference method



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History

Received 30 September 2022
Accepted 16 March 2023
Published: 26 May 2023



--------------------------------------------------------------------------------

Keywords
 * Time-dependent PDE
 * recurrent neural network
 * long short-term memory
 * physics-informed neural network
 * finite difference method

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