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

TL;DR

This work addresses solving linear and nonlinear time-fractional partial integro-differential equations (TFPIDEs) with Caputo derivatives of orders $0<\beta_k<1$ and $1<\beta_m<2$. It introduces a tensor neural network (TNN) subspace method that combines a spatiotemporal separated TNN with Gauss-Jacobi quadrature to discretize the Caputo derivative via a multiplicative factor $t^{\mu}$, enabling efficient, parallel computation and high-accuracy loss evaluations. The paper develops a Ritz-type alternating optimization framework for linear problems and a two-loop scheme for nonlinear problems, with extensive 1D-3D numerical experiments showing relative $L^2$ errors often in the $10^{-5}$ to $10^{-8}$ range and notable improvements over fPINN-based methods. This approach provides a scalable, accurate solver for memory-rich FP systems and opens avenues for high-dimensional FPDEs and other Fredholm/Volterra-type problems in applied sciences.

Abstract

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.

Figures (13)

• Figure 1: Architecture of spatiotemporal-separated TNN. Black arrows mean linear transformation (or affine transformation). Each ending node of blue arrows is obtained by taking the scalar multiplication of all starting nodes of blue arrows that end in this ending node. The final output of TNN is derived from the summation of all starting nodes of red arrows.
• Figure 2: The relative $L^2$ error during the training process for solving \ref{['eq:multi_diffusion_wave_equations']} with $m = 1$, $\beta \in(0,1)\cup(1,2)$ and $u=(t^{\alpha_{1}}+t^{\alpha_{2}}) \sin(2\pi x)$.
• Figure 3: Numerical results of equation \ref{['eq:multi_diffusion_wave_equations']} with $m = 1$, $u=(t^{\alpha_{1}}+t^{\alpha_{2}}) \sin(2\pi x)$ for $\alpha_{1} = 0.20, \alpha_{2} = 0.30$, and $\beta =0.01$ (top row); $\alpha_{1} = 0.01,\alpha_{2} = 0.01$, and $\beta =0.5$ (middle row); $\alpha_{1} = 1.40,\alpha_{2} = 2.0$, and $\beta =2.4$ (bottom row): the exact solutions (left column), the predictions (middle column), and the absolute errors of approximation (right column).
• Figure 4: Numerical results of equation \ref{['eq:multi_diffusion_wave_equations']} with $m = 1$ and $u=(t^{\beta}+1) \sin(6\pi x)$ for $\mu=\beta = 0.70$ (top row) and $u=(t^{\beta}+1) \sin(4\pi x)$ for $\mu = \beta = 1.20$ (bottom row): the exact solutions (left column), the predictions (middle column) and the absolute errors of approximation (right column).
• Figure 5: The relative $L^2$ error during the training process for solving \ref{['eq:multi_diffusion_wave_equations']} with $m =1$, $u=(t^{\beta}+1) \sin(6\pi x)$ and $u=(t^{\beta}+1) \sin(4\pi x)$.

Theorems & Definitions (3)

• Theorem 2.1
• Remark 2.1
• Remark 4.1