RTC-fPINNs: Respecting temporal causality fPINNs method for time fractional fourth order partial differential equations

Abstract

The accuracy of conventional fractional physics-informed neural networks (fPINNs) approach suffers dramatically for time fractional partial differential equations (PDEs) with high order derivative and strong nonlinearity. In this paper, a new respecting temporal causality fPINNs (RTC-fPINNs) method for solving time fractional fourth order PDEs is studied. To start with, an auxiliary function is introduced to transform the objective equations into two second order physical systems. Next, the Caputo fractional derivative is approximated through the L1 scheme on graded mesh for resolving the solution’s initial singularity, and correspondingly, the integer order derivatives are obtained via leveraging automatic differentiation of neural networks. Then, the reformulation loss functions corresponding to soft and hard constraints of boundary conditions are adopted to respect the intrinsic causal structure of the physical systems when training the neural network models. Finally, various numerical scenarios, including time fractional fourth order subdiffusion equation, time fractional Kuramoto–Sivashinsky and Cahn–Hilliard equations, are conducted to validate the remarkable effectiveness and robustness of the RTC-fPINNs method. It is noteworthy that, when using the same parameter values, the error accuracy of the established RTC-fPINNs approach is significantly better than that of the conventional fPINNs method.