用神经网络改进随机求解椭圆方程，提速省资源
WoSNN: Stochastic Solver for PDEs with Machine Learning
- 结合神经网络与走球法，实现无网格快速求解
- 误差降75%，样本量仅需原方法8%，计算更高效
- 适合静态区域的频繁预测，训练后可瞬时输出
求解椭圆型偏微分方程是科学与工程研究中的基础步骤。经典的随机求解方法——走球法（Walk-on-Spheres, WoS）具有高精度和对不规则区域鲁棒的优点，能有效提供局部解估计。本文将机器学习技术与WoS及空间离散化方法结合，提出新型随机求解器WoS-NN，用于求解具有狄利克雷边界条件的椭圆问题，实现精确且快速的全局解与梯度近似。该方法继承了原始WoS的无网格特性与强鲁棒性。通过引入神经网络，训练完成后可在无需重采样的情况下即时完成局部预测，特别适用于静态区域内高频请求场景。实验表明，相较于传统WoS方法，新方法在仅使用8%路径样本的情况下，将误差降低约75%，显著节省计算时间与资源消耗。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。