用张量神经网络求解分数阶微分方程，精度高且适用范围广。
Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Network
- 通过张量神经网络结合高斯-雅可比积分，构建时间分数阶导数的通用数值方法。
- 在(0,1)和(1,2)阶范围内，对线性与非线性方程均实现高精度求解。
- 适合需高效求解分数阶微分方程的研究者，尤其在科学计算与工程建模中应用广泛。
本文提出一种基于自适应张量神经网络子空间的新型机器学习方法，用于求解线性时间分数阶扩散波方程及非线性时间分数阶偏积分微分方程。该框架将张量神经网络与高斯-雅可比求积法有效结合，构建了适用于时间分数阶导数（阶次范围 $ (0,1)$ 和 $(1,2)$）的通用数值方案。为高效利用高斯-雅可比求积离散Caputo导数，设计了乘以 $ t^μ $ 函数的张量神经网络，其中指数 $ μ $ 根据方程参数自适应选择。通过多个数值实验验证了所提方法在效率与精度上的优越性。
原文摘要 · Abstract (English)
In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In this framework, the tensor neural network and Gauss-Jacobi quadrature are effectively combined to construct a universal numerical scheme for the temporal Caputo derivative with orders spanning $ (0,1)$ and $(1,2)$. Specifically, in order to effectively utilize Gauss-Jacobi quadrature to discretize Caputo derivatives, we design the tensor neural network function multiplied by the function $t^μ$ where the power $μ$ is selected according to the parameters of the equations at hand. Finally, some numerical examples are provided to validate the efficiency and accuracy of the proposed tensor neural network based machine learning method.
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