Abstract In this paper, neural networks based on Legendre polynomials are established to solve space and time fractional diffusion equations. The error functions containing adjustable parameters (the weights) for the training sets are constructed. The range of learning rate is analyzed to ensure that the error decreases with respect to training times. Several numerical examples including numerical results and graphs are illustrated. The results show that more training can achieve high precision.

中文翻译：

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求解分数阶扩散方程的神经网络方法

摘要 本文建立了基于勒让德多项式的神经网络求解空间和时间分数阶扩散方程。构建包含训练集可调参数（权重）的误差函数。分析学习率的范围以确保误差相对于训练时间减少。说明了几个数值示例，包括数值结果和图表。结果表明，更多的训练可以达到较高的精度。