# WoSNN: Stochastic Solver for PDEs with Machine Learning

arXiv: https://arxiv.org/abs/2509.00204 · HTML: https://mlsanity.com/paper/2509.00204/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 29 Aug 2025.

## Builds on (most cited first, 6 of 6)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients](https://mlsanity.com/paper/2201.13240/) (2022)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)

## Abstract (arXiv)

Solving elliptic partial differential equations (PDEs) is a fundamental step in various scientific and engineering studies. As a classic stochastic solver, the Walk-on-Spheres (WoS) method is a well-established and efficient algorithm that provides accurate local estimates for PDEs. In this paper, by integrating machine learning techniques with WoS and space discretization approaches, we develop a novel stochastic solver, WoS-NN. This new method solves elliptic problems with Dirichlet boundary conditions, facilitating precise and rapid global solutions and gradient approximations. The method inherits excellent characteristics from the original WoS method, such as being meshless and robust to irregular regions. By integrating neural networks, WoS-NN also gives instant local predictions after training without re-sampling, which is especially suitable for intense requests on a static region. A typical experimental result demonstrates that the proposed WoS-NN method provides accurate field estimations, reducing errors by around 75% while using only 8% of path samples compared to the conventional WoS method, which saves abundant computational time and resource consumption.