Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order
Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order
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DOI：
10.1016/j.chaos.2022.111856
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
Haidong Qu;Xuan Liu;Xin Lu;Mati ur Rahman;Zihang She
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文献类型：
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作者：
Haidong Qu;Xuan Liu;Xin Lu;Mati ur Rahman;Zihang She
在本文中，提出了神经网络方法（NNM）来求解具有非线性源项的时空变阶分数平流扩散方程。该网络是通过使用具有可调系数的移位勒让德正交多项式建立的。根据变分数阶导数的性质，从理论上推导了神经网络的损失函数。假设源函数满足Lipschitz假设，详细讨论了学习率的合理范围。然后在训练集上重复训练神经网络，以减少两个数值示例的损失函数。数值结果表明，神经网络方法在解决一些非线性变量分数阶问题时优于有限差分法。最后，给出了几张图表和一些数值分析来证实我们的结论。
In this article, neural network method (NNM) is presented to solve the spatiotemporal variable-order fractional advection-diffusion equation with a nonlinear source term. The network is established by using shifted Legendre orthogonal polynomials with adjustable coefficients. According to the properties of variable fractional derivative, the loss function of neural network is deduced theoretically. Assume that the source function satisfies the Lipschitz hypothesis, the reasonable range for learning rate is discussed in details. Then neural networks are trained repeatedly on the training set to reduce the loss functions for two numerical examples. Numerical results show that the neural network method is better than the finite difference method in solving some nonlinear variable fractional order problems. Finally, several graphs and some numerical analysis are given to confirm our conclusions.