基于组合神经网络的时间分数阶扩散方程计算方法

A Combined Artificial Neural Network Method for Solving Time Fractional Diffusion Equations

该文首次采用一种组合神经网络的方法,求解了一维时间分数阶扩散方程.组合神经网络是由径向基函数(RBF)神经网络与幂激励前向神经网络相结合所构造出的一种新型网络结构.首先,利用该网络结构构造出符合时间分数阶扩散方程条件的数值求解格式,同时设置误差函数,使原问题转化为求解误差函数极小值问题;然后,结合神经网络模型中的梯度下降学习算法进行循环迭代,从而获得神经网络的最优权值以及各项最优参数,最终得到问题的数值解.数值算例验证了该方法的可行性、有效性和数值精度.该文工作为时间分数阶扩散方程的求解开辟了一条新的途径.

A combined artificial neural network method was proposed to solve 1D time fractional diffusion equations. The combined artificial neural network is a new network structure constructed through combination of the radial basis function (RBF) neural network and the power series feed-forward neural network. First, the proposed new model was applied to create a numerical solution conforming to the conditions of the time fractional diffusion equation. Meanwhile, an error function was defined and the original differential equation was transformed into a minimization problem. Afterwards, the gradient descent learning algorithm was used to obtain the optimal weights of the neural network and other optimal parameters. Finally, a numerical example was given to illustrate the validity of the proposed method. The work makes a new way to the solution of 1D time fractional diffusion equations.