基于内嵌物理信息神经网络求解空间分数阶扩散方程

Solving spatial fractional diffusion equation based on embedded physical information neural network

利用内嵌物理信息神经网络方法(PINN)求解一类具有分数拉普拉斯算子的空间分数阶扩散方程,获得分数阶偏微分方程的数值解。首先将分数阶导数项采用有限差分离散算子后嵌入PINN进行求解,并借助自动微分技术进行求导;然后建立了训练误差函数,并给出方程初边值问题的相关算法,分析了神经网络的学习速率和数值误差;其次,给出数值例子,验证了用该方法求解空间分数阶扩散方程的有效性。

In this paper,the embedded physical information neural network method(PINN)is used to solve a class of spatial fractional diffusion equations with fractional Laplacian operator,and the numerical solution of the fractional partial differential equation is obtained.Firstly,the fractional derivative term is solved by using the finite difference discrete operator and embedded in PINN.Then the training error function is established,the correlation algorithm of the initial boundary value problem is given,and the learning rate and numerical error of the neural network are analyzed.Finally,a numerical example is given to verify the effectiveness of the proposed method in solving the space fractional diffusion equation.