A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machines
This work addresses the training time bottleneck for ELMs in solving partial differential equations, offering an incremental improvement over existing domain decomposition methods.
The paper tackles the computational expense of extreme learning machines (ELMs) in solving large least squares problems for high accuracy by constructing a coarse space and using a Neumann-Neumann acceleration within domain decomposition methods, resulting in significant speedup compared to previous DDM approaches for ELMs.
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.