Deep-Picard Iteration for Space-time Fractional Diffusion PDEs

TL;DR

This paper introduces a neural network-based iterative framework for solving high-dimensional space-time fractional diffusion PDEs using Monte Carlo methods and a fixed-point formulation.

Contribution

It presents a novel Deep-Picard iteration approach that avoids direct discretization of fractional operators by leveraging stochastic simulation and neural network regression.

Findings

Demonstrates stable convergence of the method.

Achieves accurate solutions in high dimensions up to d=100.

Validates effectiveness through numerical experiments.

Abstract

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 L\'evy 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.