In this paper, numerical methods for solving fractional differential equations by using a triangle neural network are proposed. The fractional derivative is considered Caputo type. The fractional derivative of the triangle neural network is analyzed first. Then, based on the technique of minimizing the loss function of the neural network, the proposed numerical methods reduce the fractional differential equation into a gradient descent problem or the quadratic optimization problem. By using the gradient descent process or the quadratic optimization process, the numerical solution to the FDEs can be obtained. The efficiency and accuracy of the presented methods are shown by some numerical examples. Numerical tests show that this approach is easy to implement and accurate when applied to many types of FDEs.

中文翻译：

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用三角神经网络求解分数阶微分方程

提出了利用三角神经网络求解分数阶微分方程的数值方法。分数导数被认为是Caputo类型。首先分析三角神经网络的分数导数。然后，基于使神经网络的损失函数最小化的技术，所提出的数值方法将分数阶微分方程式简化为梯度下降问题或二次优化问题。通过使用梯度下降过程或二次优化过程，可以得到FDE的数值解。数值算例表明了所提出方法的效率和准确性。数值测试表明，这种方法在应用于多种类型的FDE时易于实现且准确。