fTNN: a tensor neural network for fractional PDEs

2026-06-25

AI Summary

This work addresses the challenge of solving partial differential equations involving the fractional Laplacian on bounded domains, particularly in regimes characterized by strong boundary singularities and long-time simulations. The authors propose a deterministic tensor neural network framework that integrates a geometry-adaptive near-field integral decomposition, trial functions informed by boundary singularity profiles, an automated strategy for selecting dominant singular exponents, and a separable spatiotemporal network architecture trained via alternating subspace optimization. Singular and regular components are efficiently handled through Gauss–Jacobi quadrature and deterministic angular integration, while residual low-rank decomposition enhances numerical stability and accuracy. Benchmark comparisons demonstrate that the proposed method significantly outperforms fractional physics-informed neural networks (fPINNs) and Monte Carlo approaches in both high-singularity and long-time settings.

Abstract

We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field. Then the singular radial integrals are treated by Gauss-Jacobi quadrature, the regular radial integrals by Gauss quadrature, and the angular variables by deterministic angular quadrature, yielding a fully deterministic integration framework of the fractional Laplacian operator. To accurately resolve low-regularity solutions and the associated loss functional, we construct boundary-singularity-aware trial functions enriched with explicit boundary features, and propose two strategies for automatically selecting the leading exponent and evaluating the loss function from the singularity structure induced by the fractional operator, or jointly by the fractional operator and the source term. For time-dependent fractional PDEs, we design a spatiotemporally separable neural network that factorizes the time-space residual into a sum of low-dimensional temporal and spatial integrals, and we integrate this representation with an alternating neural network subspace optimization strategy for efficient training. Numerical experiments show that the proposed framework attains high accuracy on the tested benchmarks and improves substantially over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.

Problem

• fractional PDEs
• fractional Laplacian
• boundary singularities
• low-regularity solutions
• time-dependent problems

Innovation

• tensor neural network
• fractional PDEs
• deterministic quadrature
• boundary singularity
• spatiotemporal separation