A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machines

2025-03-13

AI Summary

Training extreme learning machines (ELMs) for partial differential equations (PDEs) suffers from low efficiency due to large-scale least-squares optimization. Method: We propose a novel domain decomposition framework incorporating a coarse space, specifically designed for ELMs. It introduces a geometry–algebra hybrid coarse space, hierarchical interface variable partitioning, and selective elimination to construct a Schur complement system embedding the coarse problem; a tailored Neumann–Neumann-type acceleration mechanism ensures global consistency across subdomains. Contribution/Results: This work overcomes the longstanding challenge of enforcing global consistency in domain decomposition methods applied to ELMs. Experiments on multiple PDE benchmarks demonstrate that our method accelerates training by over 50% on average compared to existing domain decomposition approaches, while preserving high solution accuracy. The framework establishes an efficient, scalable paradigm for large-scale physics-informed ELM training.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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.

Problem

• Accelerates training of Extreme Learning Machines (ELMs) using domain decomposition.
• Introduces a coarse space to reduce computational cost in large least squares problems.
• Utilizes Neumann-Neumann acceleration for improved performance in solving partial differential equations.

Innovation

• Coarse space construction for ELM acceleration
• Neumann-Neumann acceleration with coarse space
• Schur complement system for selective elimination