arXiv subjects

Qingkui Ma

Publications and source records attributed to Qingkui Ma.

3 recordsLinked to original sources

arXiv2026

fTNN: a tensor neural network for fractional PDEs

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.

cs.LG↗

arXiv2026

Quadrature-Enhanced Monte Carlo fPINN Method for High-Dimensional Fractional PDEs

Fractional PDEs involving the fractional Laplacian on bounded domains are challenging because of hypersingular nonlocal kernels, exterior Dirichlet constraints, reduced boundary regularity, and the high computational cost in high dimensions. To address these issues, we first adopt a spatially varying radius with directional distance-to-boundary information, which yields a geometry-adaptive three-part decomposition of the fractional Laplacian: singular near-field, regular interior far-field, and analytical exterior far-field contributions. Then we employ Gauss-Jacobi quadrature for the singular radial integral, Gauss quadrature for the regular interior radial integral, and Monte Carlo sampling for the angular variables. A feature-enhanced physics-informed neural network trial space is finally used to tackle the low-regularity behavior near the boundary. Through the above steps, we obtain a quadrature-enhanced Monte Carlo fractional physics-informed neural network (QE-MC-fPINN) method. Numerical experiments on fractional Poisson equations and time-dependent fractional PDEs show that, on the tested benchmarks, the proposed method outperforms two representative MC-fPINN discretizations in accuracy and convergence, especially for solutions with strong boundary singularities.

math.NA↗

arXiv2025

Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Network

In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In this framework, the tensor neural network and Gauss-Jacobi quadrature are effectively combined to construct a universal numerical scheme for the temporal Caputo derivative with orders spanning $ (0,1)$ and $(1,2)$. Specifically, in order to effectively utilize Gauss-Jacobi quadrature to discretize Caputo derivatives, we design the tensor neural network function multiplied by the function $t^{\mu}$ where the power $\mu$ is selected according to the parameters of the equations at hand. Finally, some numerical examples are provided to validate the efficiency and accuracy of the proposed tensor neural network based machine learning method.

cs.LG↗