WoSNN: Stochastic Solver for PDEs with Machine Learning
This work addresses the computational inefficiency in solving PDEs for scientific and engineering applications, offering a faster and more accurate meshless solver, though it is incremental as it builds on existing stochastic methods.
The paper tackled solving elliptic PDEs with Dirichlet boundary conditions by integrating machine learning with the Walk-on-Spheres method, resulting in a new solver (WoS-NN) that reduces errors by about 75% and uses only 8% of path samples compared to conventional methods.
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.