A Physics-Informed Recurrent Neural Network for Solving Time-Dependent Partial Differential Equations

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.