为极限学习机设计粗空间加速方法，显著提升求解效率。
A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machines
- 通过划分界面变量构建粗空间，引入选择性消去生成带粗问题的舒尔补系统。
- 采用内曼-内曼加速策略，数值实验显示比先前方法提速明显。
- 适合需要快速求解高精度偏微分方程的科研与工程计算场景。
极限学习机（ELM）通过预设隐藏层参数并用最小二乘法求解输出层系数，可比物理信息神经网络更快速、更准确地求解偏微分方程。然而，当高精度需求导致大规模最小二乘问题时，其计算成本仍较高。针对此问题，本文为ELM构建了粗空间，进一步加速训练过程。通过将界面变量划分为粗变量与非粗变量，实施选择性消去，得到仅含非粗变量的舒尔补系统，并将粗问题嵌入其中。所提方法的关键在于利用粗空间的内曼-内曼加速机制。数值实验表明，该方法相比此前针对ELM的域分解方法具有显著加速效果。
原文摘要 · Abstract (English)
Extreme learning machines (ELMs), which preset hidden layer parameters and solve for last layer coefficients via a least squares method, can typically solve partial differential equations faster and more accurately than Physics Informed Neural Networks. However, they remain computationally expensive when high accuracy requires large least squares problems to be solved. Domain decomposition methods (DDMs) for ELMs have allowed parallel computation to reduce training times of large systems. This paper constructs a coarse space for ELMs, which enables further acceleration of their training. By partitioning interface variables into coarse and non-coarse variables, selective elimination introduces a Schur complement system on the non-coarse variables with the coarse problem embedded. Key to the performance of the proposed method is a Neumann-Neumann acceleration that utilizes the coarse space. Numerical experiments demonstrate significant speedup compared to a previous DDM method for ELMs.
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