fTNN: a tensor neural network for fractional PDEs

2026-06-25 · Source: Takara TLDR - Daily AI Papers · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, medium

Summary

The fTNN is a deterministic tensor neural network subspace method designed to solve problems involving the fractional Laplacian on bounded domains, exemplified by the fractional Poisson equation and time-dependent fractional advection-diffusion equation. This framework employs a geometry-adapted integration split with a spatially dependent near-field radius, decomposing the fractional Laplacian into singular near-field, regular interior far-field, and analytical exterior far-field contributions. Singular radial integrals are processed by Gauss-Jacobi quadrature, regular radial integrals by Gauss quadrature, and angular variables by deterministic angular quadrature, forming a fully deterministic integration. To handle low-regularity solutions and loss functions, the fTNN uses boundary-singularity-aware trial functions and strategies for exponent selection and loss evaluation based on singularity structure. For time-dependent PDEs, it features a spatiotemporally separable neural network and an alternating optimization strategy. Numerical experiments confirm high accuracy and substantial improvements over fPINN and Monte Carlo baselines, especially for strong boundary singularities and long-time simulations.

Key takeaway

For research scientists and engineers developing solutions for fractional partial differential equations, particularly those with strong boundary singularities or requiring long-time simulations, the fTNN offers a robust, high-accuracy alternative. You should consider its deterministic integration framework and boundary-singularity-aware trial functions to achieve superior solution fidelity. This method significantly outperforms existing fPINN and Monte Carlo baselines, providing a more reliable approach for complex fractional calculus problems.

Key insights

fTNN is a deterministic tensor neural network achieving high accuracy for fractional PDEs, outperforming fPINN and Monte Carlo.

Principles

• Decompose fractional Laplacian into singular near-field, regular interior far-field, and analytical exterior far-field.
• Employ boundary-singularity-aware trial functions for low-regularity solutions.

Method

The fTNN uses a geometry-adapted integration split with specific quadrature methods, boundary-singularity-aware trial functions, and a spatiotemporally separable neural network with alternating optimization.

In practice

• Apply geometry-adapted integration split for fractional Laplacian problems.
• Utilize boundary-singularity-aware trial functions for low-regularity solutions.

Topics

• Fractional PDEs
• Tensor Neural Networks
• Fractional Laplacian
• Numerical Integration
• Boundary Singularities
• Scientific Machine Learning

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.