低保真FDM谱引导的神经特征值求解器

中文摘要 AI 辅助

算子特征值问题在科学领域普遍存在。经典方法通常将算子离散化为矩阵，然后求解由此产生的矩阵特征值问题。这在低维情况下效果良好，但随着维度增长，细网格在内存和计算上迅速变得昂贵。基于神经网络的求解器避免了存储这些大网格，但近期最先进的神经方法可能需要数十万步训练，并且可能难以找到所需的特征值。我们表明这两种方法可以相互帮助。粗网格有限差分法（FDM）计算作为算子谱的廉价数值模型。我们将近似特征值用作神经求解器训练期间的固定移位，因为它们只需定位谱的相关部分。我们还引入了稳定逆幂法神经网络（SIPMNN），这是一种针对高维问题的更稳定的训练过程。在$d=10$的五个测试问题上，组合方法在整体上比测试的纯神经替代方法更准确，同时使用的迭代次数减少了八到十倍。

英文摘要

Operator eigenvalue problems appear throughout science. Classical methods usually discretize the operator into a matrix and then solve the resulting matrix eigenvalue problem. This works well in low dimensions, but fine grids quickly become expensive in both memory and computation as the dimension grows. Neural network based solvers avoid storing these large grids, but recent state of the art neural methods can require hundreds of thousands of training steps and may struggle to find the desired eigenvalues. We show that the two approaches can help each other. A coarse finite difference method (FDM) calculation acts as a cheap numerical model of the operator spectrum. We use the approximate eigenvalues as fixed shifts during the training of the neural solver, as they only need to locate the relevant part of the spectrum. We also introduce Stabilized Inverse Power Method Neural Network (SIPMNN), a more stable training procedure for higher-dimensional problems. Across five test problems at $d=10$, the combined approach is more accurate overall than the tested fully neural alternatives while using eight to ten times fewer iterations.

发表机构

• BITS Pilani（比拉尼理工学院）
• IBM Research（IBM研究院）

机构由 AI 辅助整理，请以论文原文为准。