arXiv subjects

Byungeun Ryoo

Publications and source records attributed to Byungeun Ryoo.

3 recordsLinked to original sources

arXiv2026

A Fully Dual Domain Decomposition Method for Extreme Learning Machines Using GenEO Spectral Coarse Spaces

Extreme learning machines (ELMs) are a class of machine learning algorithms that randomly intialize and subsequently fix the hidden layer parameters, and solve for the last layer coefficients using a least squares method. ELMs have been successfully applied to the solution of various partial differential equations, but the cost of solving large least squares problems limits their use in large scale problems or when high accuracy is desired. To address this limitation, domain decomposition methods (DDMs) have been developed for ELMs, reducing training times via parallel computation. Yet these methods, even those incorporating a coarse space, have struggled with scalability. This paper introduces a novel nonoverlapping DDM for ELMs that treats and enforces continuity and flux conditions equally. Additive Schwarz preconditioners and GenEO (Generalized Eigenvalues problems in the Overlaps) coarse spaces are constructed to accelerate the method. Numerical experiments show the new method outperforms previous DDMs for ELMs in both iteration count and accuracy, especially for the Stokes and wave equations. Furthermore, weak scalability has been observed in up to 4,096 subdomains when employing GenEO coarse spaces.

math.NA↗

arXiv2025

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

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.

math.NA↗

arXiv2024

A Nonoverlapping Domain Decomposition Method for Extreme Learning Machines: Elliptic Problems

Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known to be much faster than Physics informed neural networks. However, classical ELM is still computationally expensive when a high level of representation is desired in the solution as this requires solving a large least squares system. In this paper, we propose a nonoverlapping domain decomposition method (DDM) for ELMs that not only reduces the training time of ELMs, but is also suitable for parallel computation. In numerical analysis, DDMs have been widely studied to reduce the time to obtain finite element solutions for elliptic PDEs through parallel computation. Among these approaches, nonoverlapping DDMs are attracting the most attention. Motivated by these methods, we introduce local neural networks, which are valid only at corresponding subdomains, and an auxiliary variable at the interface. We construct a system on the variable and the parameters of local neural networks. A Schur complement system on the interface can be derived by eliminating the parameters of the output layer. The auxiliary variable is then directly obtained by solving the reduced system after which the parameters for each local neural network are solved in parallel. A method for initializing the hidden layer parameters suitable for high approximation quality in large systems is also proposed. Numerical results that verify the acceleration performance of the proposed method with respect to the number of subdomains are presented.

math.NA↗