论文
arXiv
GeoSimulation
中文标题
求解时空分数阶扩散偏微分方程的深度Picard迭代法
English Title
Deep-Picard Iteration for Space-time Fractional Diffusion PDEs
Zhijun Zeng, Zhitong Chen, Ling Qin, Yi Zhu
发布时间
2026/5/1 14:41:36
来源类型
preprint
语言
en
摘要
中文对照

本文提出一种面向高维非线性时空分数阶扩散方程的深度Picard迭代框架。该方法基于非线性分数阶Feynman–Kac不动点公式,通过蒙特卡洛模拟对应的分数阶动力学过程,替代对Caputo记忆项及非局部分数阶拉普拉斯算子的直接离散。每次Picard更新均通过随机标签生成实现,并借助有监督神经网络回归完成,从而避免涉及分数阶微分算子的残差最小化。分数阶轨迹由离散化的beta稳定次序器与旋转对称alpha稳定Lévy过程的球面游走(walk-on-spheres)型模拟耦合生成。二维及高维测试问题的数值实验表明该方法具有稳定的Picard收敛性与高精度近似能力,所报告测试维度最高达d=100。

English Original

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.

元数据
arXiv2605.00456v1
来源arXiv
类型论文
抽取状态raw
关键词
GeoSimulation
math.NA
math.PR