Deep-Picard Iteration for Space-time Fractional Diffusion PDEs
It provides a scalable method for high-dimensional fractional PDEs, which are challenging for traditional numerical methods.
The paper introduces a Deep-Picard iteration framework for solving high-dimensional nonlinear space-time fractional diffusion PDEs, achieving stable convergence and accurate approximation in tests up to dimension 100.
We propose a Deep-Picard iteration framework for high-dimensional nonlinear space-time fractional diffusion equations.The method is based on a nonlinear fractional Feynman--Kac fixed-point formulation, which replaces direct discretization of the Caputo memory term and the nonlocal fractional Laplacian by Monte Carlo simulation of the associated fractional dynamics. Each Picard update is approximated by stochastic label generation and realized through supervised neural-network regression, thereby avoiding residual minimization involving fractional differential operators. The fractional trajectories are generated by coupling a discretized beta-stable subordinator with a walk-on-spheres-type simulation of the rotationally symmetric alpha-stable Lévy process. Numerical experiments on two-dimensional and high-dimensional test problems ddemonstrate stable Picard convergence and accurate approximation, with tests reported up to dimension d=100.