Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Networks

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 $t^{\mu}$ where the power $\mu$ 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.

@article{lin2025_2504.01440,
  title={ Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Networks },
  author={ Zhongshuo Lin and Qingkui Ma and Hehu Xie and Xiaobo Yin },
  journal={arXiv preprint arXiv:2504.01440},
  year={ 2025 }
}