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

Abstract

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.