FFNN: Fractional order basis function multi-step neural network method for fractional partial differential equations

Abstract

With the advancement in artificial intelligence technology, the increasing number of researchers utilize it to address complex equations in ocean engineering. So the technology of artificial intelligence has become a practical area of research. In this paper, we design a novel method to solve the fractional order long water wave equation, which is called the fractional order basis function multi-step neural network. Firstly, a power series is constructed based on a fractional order basis function, which serves as the approximate solution. Secondly, neural networks and the initial conditions of differential equations are integrated into the construction of approximate solutions. Furthermore, the solution is discretized, and a multi-step unfolding strategy is employed on the resulting discrete solution. This approach ensures that each point in the solution is influenced by its predecessor. By means of repeated applications of the optimization algorithm, the residuals are successively diminished, thereby yielding approximate solutions to the equations. Finally, the efficacy and versatility of the proposed strategy were validated through a series of numerical experiments. Compared with the method of fractional physics-informed neural networks, there are up to $18.7$-fold and $22.8$-fold increases in stability of average and maximum residuals. Simultaneously, initial conditions are retained in new solutions.