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

2025-04-02

AI Summary

This work addresses the numerical solution of time-fractional diffusion-wave equations (with order α ∈ (0,1) ∪ (1,2)) and nonlinear fractional partial integro-differential equations. We propose a machine learning method based on an adaptive tensor neural network subspace. Our approach innovatively couples tensor neural networks with Gauss–Jacobi orthogonal bases and incorporates a t^μ-weighted architecture to accurately capture the singular temporal behavior induced by Caputo fractional derivatives. By integrating tensor low-rank approximation, high-precision Gauss–Jacobi quadrature, and adaptive power-function weighting, we achieve a robust, general-purpose, high-order discretization scheme. Numerical experiments demonstrate that the method delivers high accuracy (errors of 10⁻⁴–10⁻⁶) and strong generalization across diverse linear and nonlinear problems. It significantly outperforms conventional finite difference and spectral methods—particularly in resolving initial singularities—achieving over one order-of-magnitude improvement in computational efficiency.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Diffusion Models for Vision
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Machine learning and data science for the Web

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 $t^{mu}$ 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.

Problem

• Solving time-fractional diffusion-wave and integro-differential equations
• Combining tensor neural networks with Gauss-Jacobi quadrature
• Validating accuracy of adaptive tensor neural network method

Innovation

• Adaptive tensor neural network subspace method
• Combines tensor neural networks with Gauss-Jacobi quadrature
• Designs t^μ multiplied tensor neural network functions