fTNN: a tensor neural network for fractional PDEs

Numerical solutions for fractional partial differential equations (fPDEs) on bounded domains are notoriously difficult due to the nonlocal nature of the fractional Laplacian, which leads to dense matrices and reduced solution regularity at boundaries. Existing physics-informed neural network (PINN) methods often rely on Monte Carlo sampling to approximate these nonlocal operators, which introduces stochastic noise that limits accuracy. This paper asks whether a fully deterministic integration framework can be constructed to improve accuracy and efficiency for these problems.

The authors introduce the fTNN (Tensor Neural Network) method, which replaces stochastic angular sampling with deterministic quadrature. The framework employs a geometry-adaptive decomposition of the fractional Laplacian into three distinct components: a singular near-field, a regular interior far-field, and an analytical exterior far-field. Radial integrals are evaluated using Gauss-Jacobi or Gauss quadrature, while angular integrals are computed via deterministic quadrature rules. To handle low-regularity solutions, the authors construct trial functions enriched with explicit boundary features, using adaptive strategies to select the leading singularity exponent. For time-dependent problems, they design a spatiotemporally separable neural network (STSNN) that factorizes the residual into low-dimensional temporal and spatial integrals, allowing for efficient training via alternating subspace optimization.

Numerical experiments demonstrate that the fTNN framework achieves significantly higher accuracy than existing fPINN and Monte Carlo-based baselines. The deterministic nature of the integration eliminates the variance associated with stochastic sampling, leading to more stable and precise results. The STSNN architecture is particularly effective for long-time simulations and problems with strong boundary singularities, as it allows for dense temporal quadrature without the memory explosion typically associated with nonlocal Caputo derivatives.