fTNN: a tensor neural network for fractional PDEs

2026-06-25 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Scientific Computing & Numerical Methods · Depth: Expert, quick

Summary

The fTNN is a deterministic tensor neural network subspace method designed to solve problems involving the fractional Laplacian on bounded domains, specifically addressing the fractional Poisson equation and time-dependent fractional advection-diffusion equation. This framework employs a geometry-adapted integration split that decomposes the fractional Laplacian into singular near-field, regular interior far-field, and analytical exterior far-field contributions. It utilizes Gauss-Jacobi quadrature for singular radial integrals, Gauss quadrature for regular radial integrals, and deterministic angular quadrature for angular variables. To handle low-regularity solutions and loss functions, fTNN constructs boundary-singularity-aware trial functions and proposes strategies for automatically selecting leading exponents. For time-dependent fractional PDEs, it features a spatiotemporally separable neural network integrated with an alternating neural network subspace optimization strategy. Numerical experiments demonstrate high accuracy and substantial improvements over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.

Key takeaway

For AI Scientists or Machine Learning Engineers developing accurate solvers for fractional PDEs, the fTNN offers a deterministic, high-accuracy alternative to fPINN and Monte Carlo methods. If your work involves problems with strong boundary singularities or long-time simulations, consider fTNN's integration framework and singularity-aware functions to improve solution fidelity and training efficiency for complex fractional problems.

Key insights

fTNN is a deterministic tensor neural network method for fractional PDEs, using geometry-adapted integration and singularity-aware trial functions for high accuracy.

Principles

• Decompose fractional Laplacian into near, interior far, and exterior far fields.
• Use specific quadratures for different integral types.
• Enrich trial functions with explicit boundary features for low-regularity solutions.

Method

fTNN employs a geometry-adapted integration split, uses Gauss-Jacobi and Gauss quadratures, and deterministic angular quadrature. It constructs boundary-singularity-aware trial functions and uses a spatiotemporally separable neural network with alternating subspace optimization for time-dependent problems.

In practice

• Apply fTNN to fractional Poisson equations.
• Use fTNN for time-dependent fractional advection-diffusion.
• Improve accuracy for problems with strong boundary singularities.

Topics

• Fractional PDEs
• Tensor Neural Networks
• Fractional Laplacian
• Numerical Integration
• Boundary Singularities
• Spatiotemporal Neural Networks

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.