Research On Numerical Solution Of Fractional Order Partial Differential Equations Based On Neural Network Models

Fractional order partial differential equations(FPDEs)often appear in systems with memory effect,spatial non-locality or dense law,and play an important role in mathematics,physics,biology,mechanical engineering,control theory,finance and electronic engineering.In recent years,artificial neural network algorithms have also been widely used in various fields.Due to the high expressiveness of neural networks in function approximation,this paper will introduce the algorithm for solving FPDEs based on neural network models.We use the fractional physical-informed neural networks(FPINN)to solve the time FPDEs.FPINN use the automatic differentiation technique of integer order operators and the numerical discretization of fractional order operators to construct the residuals in the loss function,which solves the problem that the chain rule in automatic differentiation technology is not applicable to fractional operators.First,we propose the L1-2 differential format to discretize the Caputo derivative,and compare it with the classical L1 differential format in numerical experiments.Then we find that the approximate solution obtained by L1-2 differential format has higher accuracy.At the same time,we explain the observed results by analyzing the four error sources,including discrete error,sampling error,approximation error and optimization error.We can observe that the total error decays monotonically and the error can be reduced by adjusting the parameters,but finally total error will saturates due to the optimization error.For the spatial FPDEs,we first use the Galerkin method to discretize the space variable,then promote feedforward-recurrent neural network model(FNN-RNN)to solve systems of equations.For the large time interval,we divide it into small intervals and apply the FNN-RNN to solve the numerical solution on each intervals.The initial value of the first interval is known and the initial values of the rest of the intervals are taken from the final value of the previous interval,finally we can obtain the numerical solution on the entire interval.This method can solve many problems with large time range.For time-space FPDEs,we propose a recursive formula based on the time fractional derivative and generalize the FNN-RNN to the fFNN-RNN.Through experiments,we can discover increasing the spatial discretization dimension or decreasing the time step can effectively reduce the error.