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FNWoS: Fractional Neural Walk-on-Spheres Methods for High-Dimensional PDEs Driven by $α$-stable Lévy Process on Irregular Domains

Ling Guo, Mingxin Qin, Changtao Sheng, Hao Wu, Fanhai Zeng

TL;DR

The paper tackles high-dimensional fractional Poisson equations driven by $\alpha$-stable Lévy processes on irregular domains. It blends a simplified, derivative-free walk-on-spheres scheme with neural surrogates, underpinned by a Feynman-Kac representation, to amortize Monte Carlo sampling and enable fast, scalable solutions up to $d=1000$. Key contributions include a simplified FWoS formulation, a neural-accelerated FNWoS method, a truncated-path variant (BFNWoS) with buffered supervision, and data-refinement strategies; collectively enabling accurate solutions with far fewer trajectories than traditional methods. The results demonstrate strong accuracy and scalability across a range of domains and dimensions, highlighting the practical impact for nonlocal PDEs in high dimensions.

Abstract

In this paper, we develop a highly parallel and derivative-free fractional neural walk-on-spheres method (FNWoS) for solving high-dimensional fractional Poisson equations on irregular domains. We first propose a simplified fractional walk-on-spheres (FWoS) scheme that replaces the high-dimensional normalized weight integral with a constant weight and adopts a correspondingly simpler sampling density, substantially reducing per-trajectory cost. To mitigate the slow convergence of standard Monte Carlo sampling, FNWoS is then proposed via integrating this simplified FWoS estimator, derived from the Feynman-Kac representation, with a neural network surrogate. By amortizing sampling effort over the entire domain during training, FNWoS achieves more accurate evaluation at arbitrary query points with dramatically fewer trajectories than classical FWoS. To further enhance efficiency in regimes where the fractional order $α$ is close to 2 and trajectories become excessively long, we introduce a truncated path strategy with a prescribed maximum step count. Building on this, we propose a buffered supervision mechanism that caches training pairs and progressively refines their Monte Carlo targets during training, removing the need to precompute a highly accurate training set and yielding the buffered fractional neural walk-on-spheres method (BFNWoS). Extensive numerical experiments, including tests on irregular domains and problems with dimensions up to $1000$, demonstrate the accuracy, scalability, and computational efficiency of the proposed methods.

FNWoS: Fractional Neural Walk-on-Spheres Methods for High-Dimensional PDEs Driven by $α$-stable Lévy Process on Irregular Domains

TL;DR

The paper tackles high-dimensional fractional Poisson equations driven by -stable Lévy processes on irregular domains. It blends a simplified, derivative-free walk-on-spheres scheme with neural surrogates, underpinned by a Feynman-Kac representation, to amortize Monte Carlo sampling and enable fast, scalable solutions up to . Key contributions include a simplified FWoS formulation, a neural-accelerated FNWoS method, a truncated-path variant (BFNWoS) with buffered supervision, and data-refinement strategies; collectively enabling accurate solutions with far fewer trajectories than traditional methods. The results demonstrate strong accuracy and scalability across a range of domains and dimensions, highlighting the practical impact for nonlocal PDEs in high dimensions.

Abstract

In this paper, we develop a highly parallel and derivative-free fractional neural walk-on-spheres method (FNWoS) for solving high-dimensional fractional Poisson equations on irregular domains. We first propose a simplified fractional walk-on-spheres (FWoS) scheme that replaces the high-dimensional normalized weight integral with a constant weight and adopts a correspondingly simpler sampling density, substantially reducing per-trajectory cost. To mitigate the slow convergence of standard Monte Carlo sampling, FNWoS is then proposed via integrating this simplified FWoS estimator, derived from the Feynman-Kac representation, with a neural network surrogate. By amortizing sampling effort over the entire domain during training, FNWoS achieves more accurate evaluation at arbitrary query points with dramatically fewer trajectories than classical FWoS. To further enhance efficiency in regimes where the fractional order is close to 2 and trajectories become excessively long, we introduce a truncated path strategy with a prescribed maximum step count. Building on this, we propose a buffered supervision mechanism that caches training pairs and progressively refines their Monte Carlo targets during training, removing the need to precompute a highly accurate training set and yielding the buffered fractional neural walk-on-spheres method (BFNWoS). Extensive numerical experiments, including tests on irregular domains and problems with dimensions up to , demonstrate the accuracy, scalability, and computational efficiency of the proposed methods.
Paper Structure (11 sections, 4 theorems, 59 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 11 sections, 4 theorems, 59 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Simplified FWoS scheme based on Feynman-Kac formula
  3. Feynman-Kac representation in expectation form.
  4. Implementation of the simplified FWoS scheme
  5. Two fractional neural walk-on-spheres methods
  6. Fractional neural walk-on-spheres method
  7. Buffered fractional neural walk-on-spheres method
  8. Stochastic path truncation
  9. Data refinement based on neural caches
  10. Numerical Results
  11. Conclusion

Key Result

Lemma 2.1

Let $\Omega=\mathbb{B}_r^d$ with $r>0$ centered at the origin, and assume that $f \in C^{\alpha+\varepsilon}(\mathbb{B} _r^d) \cap C(\overline{\mathbb{B} _r^d})$ and $g \in L_\alpha ^1( \mathbb{R}^d\backslash \mathbb{B} _r^d)$; then the expectation form of the solution of fractional-laplace at the c where $P_r$ is defined by Pr, the normalized weight function $\omega_r:=\omega_{r}(d,\alpha)= \frac

Figures (7)

  • Figure 1: Random walk paths on a 2D irregular domain generated by the simplified FWoS.
  • Figure 2: FNWoS with the maximum number of steps $K$. Red ball: center of the inscribed ball at the $K$-th step of the red trajectory lies in the $\varepsilon$-region; Blue ball: center of the inscribed ball at the $K$-th step of the blue trajectory lies in the interior of $\Omega$.
  • Figure 3: The relative $\ell^2$-errors of FWoS, FNWoS and BFNWoS for Example 4.1 on the 10D unit ball. (a): The relative $\ell^2$-errors of FWoS versus the parameter $N$ with various $\alpha$. (b): Comparison of the relative $\ell^2$-errors of FWoS and FNWoS with respect to trajectories $N$ for $\alpha = 0.4$ and $1.2$. (c): Comparison of the relative $\ell^2$-errors of FWoS and BFNWoS with respect to the maximum number of steps $K$. The error of FWoS is shown as a horizontal line at $\alpha = 1.6$ and $1.9$. The colored lines and shaded regions correspond to mean values and one-standard-deviation bands of relative $\ell^2$-errors.
  • Figure 4: Computational time breakdown and corresponding prediction accuracy of FWoS, FNWoS, and BFNWoS for Example 4.1 on the 50D unit ball. (a) Wall-clock time averaged over five independent runs, decomposed into Monte Carlo sampling time used to evaluate the FWoS estimator or to generate supervision targets and network training time used to refine supervision targets and train the network. FWoS has no training stage, so its training time is zero. (b) Corresponding prediction accuracy measured by relative $\ell^2$-error, obtained from the same runs as in (a).
  • Figure 5: The accuracy of FNWoS for Example 4.2 on the 2D unit disk with a low-regularity solution. (a): absolute error for $\alpha = 0.5$. (b): absolute error for $\alpha = 1.5$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Lemma 2.1
  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.2
  • Corollary 2.1
  • Remark 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • ...and 1 more