# fTNN: a tensor neural network for fractional PDEs Source: [fTNN: a tensor neural network for fractional PDEs]() Collection index: [llms.txt]() Data updated: 2026-06-27T05:00:11.922Z Authors: Qingkui Ma, Hehu Xie, Xiaobo Yin Published: 2026-06-25T15:13:55.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Solving fractional Poisson equations on fPE on Unit Ball (d=2, v1.0) (test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=2, u\_exact = (1 - \|\|x\|\|^2)^(1+alpha/2)) 1.0 (test); Relative L2 Error: 0.0000022. - [Fractional Poisson Equation Solving on Unit Ball (d=3, test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=3, u\_exact = (1 - \|\|x\|\|^2)^(1+alpha/2)) 1.0 (test); Relative L2 Error (Test): 0.000356. - [Solving fractional Poisson equations on fPE on Unit Ball (d=2, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=2, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test); Relative L2 Test Error: 0.0000172. - [Solving fractional Poisson equations on fPE on Unit Ball (d=3, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=3, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test); Relative L2 Test Error: 0.000158. - [Solving Fractional Advection-Diffusion Equations on ADE 2D Case A2 (T=100)](): Solving time-space fractional Advection-Diffusion Equations; Time-space fractional ADE 2D case A2 T=100 (long-time simulation); Relative L2 Error: 0.0001889. - [Solving time-space fractional Advection-Diffusion Equations on Time-space fractional ADE 3D, case B1, T=100 (long-time simulation)](): Solving time-space fractional Advection-Diffusion Equations; Time-space fractional ADE 3D, case B1, T=100 (long-time simulation); Relative L2 test error: 0.00009486. ## Original sources and implementation [Original paper]() [PDF]()