Deep Learning Method For High-Dimensional Fractional Order Differential Equations

Fractional order calculus,as a generalization of integer order calculus,has a potential that integer order calculus does not have.In the past decades,fractional-order differential equations have been widely used in the fields of physics,biology,medicine and other sciences and engineering.Solving high-dimensional partial differential equations has always been a difficult problem in computational science,and traditional algorithms are prone to"Curse of Dimensionality" in high-dimensional conditions,i.e.,the computational complexity increases exponentially with the increase of dimensionality.In this paper,the solution of a high-dimensional fractional order differential equation containing the right Riemann-Liouville fractional order derivative,the left Caputo fractional order derivative and boundary conditions is approximated.The solution of the original equation is transformed into solving an optimal control problem by determining the variational structure of the original equation,for which a deep learningbased method,namely the deep Ritz method,is applied,which can effectively avoid the curse of dimensionality.In order to obtain a more accurate approximate solution,this paper also improves the point-taking method of the deep Ritz method.In order to verify the effectiveness of the depth Ritz method in approximating fractional order differential equations,experimental data of the depth Ritz method on three equations are given in this paper,and the results show that the method can better approximate low,medium and high dimensional fractional order differential equations,respectively.This paper also provides a theoretical proof that the solutions of such fractional order differential equations as mentioned above can be approximated by a residual network,and gives an upper bound on the total number of parameters of the residual network.The proof is mediated by the variational iteration method for fractional order differential equations.It is first shown that the solution of the original equation can be approximated by the iterative format obtained by the variational iteration method,and then is proved that this iterative format can be approximated by the residual network.