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

TL;DR

This paper introduces QE-MC-fPINN, a novel method combining quadrature, Monte Carlo sampling, and physics-informed neural networks to efficiently solve high-dimensional fractional PDEs with boundary singularities.

Contribution

The paper develops a geometry-adaptive decomposition of the fractional Laplacian and integrates specialized quadrature and neural network techniques for improved accuracy.

Findings

Outperforms existing MC-fPINN methods in accuracy and convergence.

Effective handling of boundary singularities in fractional PDEs.

Demonstrated on fractional Poisson and time-dependent 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.