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

• URL: http://arxiv.org/abs/2503.10032v1
• Date: Thu, 13 Mar 2025 04:24:55 GMT
• Title: A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machines
• Authors: Chang-Ock Lee, Byungeun Ryoo,
• Abstract summary: Extreme learning machines (ELMs) can solve partial differential equations faster and more accurately than Physics Informed Neural Networks.<n>They remain computationally expensive when high accuracy requires large least squares problems to be solved.<n>This paper constructs a coarse space for ELMs, which enables further acceleration of their training.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Enabling Automatic Differentiation with Mollified Graph Neural Operators [75.3183193262225]
  We propose the mollified graph neural operator (mGNO), the first method to leverage automatic differentiation and compute emphexact gradients on arbitrary geometries. For a PDE example on regular grids, mGNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences. It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough.
  arXiv  Detail & Related papers  (2025-04-11T06:16:30Z)
• MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation [48.41289705783405]
  We propose a PDE-embedded network with multiscale time stepping (MultiPDENet)<n>In particular, we design a convolutional filter based on the structure of finite difference with a small number of parameters to optimize.<n>A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction.
  arXiv  Detail & Related papers  (2025-01-27T12:15:51Z)
• A Nonoverlapping Domain Decomposition Method for Extreme Learning Machines: Elliptic Problems [0.0]
  Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. 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.
  arXiv  Detail & Related papers  (2024-06-22T23:25:54Z)
• A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs [0.0]
  We introduce a novel Deep Learning algorithm that minimizes the computational cost of the Algebraic multigrid method when used as a finite element solver. We experimentally prove that the pooling successfully reduces the computational cost of processing a large sparse matrix and preserves the features needed for the regression task at hand.
  arXiv  Detail & Related papers  (2023-04-21T09:18:56Z)
• Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems [64.29491112653905]
  We propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG with the denoised data ensures the data consistency update to remain in the tangent space. Our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.
  arXiv  Detail & Related papers  (2023-03-10T07:42:49Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers.<n>We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.<n>Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• Numerical Approximation in CFD Problems Using Physics Informed Machine Learning [0.0]
  The thesis focuses on various techniques to find an alternate approximation method that could be universally used for a wide range of CFD problems. The focus stays over physics informed machine learning techniques where solving differential equations is possible without any training with computed data. The extreme learning machine (ELM) is a very fast neural network algorithm at the cost of tunable parameters.
  arXiv  Detail & Related papers  (2021-11-01T22:54:51Z)
• On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements [0.0]
  We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE) In ELM the hidden-layer coefficients in the neural network are assigned to random values generated on $[-R_m,R_m]$ and fixed. We present a method for computing the optimal value of $R_m$ based on the differential evolution algorithm.
  arXiv  Detail & Related papers  (2021-10-27T02:05:26Z)
• Research of Damped Newton Stochastic Gradient Descent Method for Neural Network Training [6.231508838034926]
  First-order methods like gradient descent(SGD) are recently the popular optimization method to train deep neural networks (DNNs) In this paper, we propose the Damped Newton Descent(DN-SGD) and Gradient Descent Damped Newton(SGD-DN) methods to train DNNs for regression problems with Mean Square Error(MSE) and classification problems with Cross-Entropy Loss(CEL) Our methods just accurately compute a small part of the parameters, which greatly reduces the computational cost and makes the learning process much faster and more accurate than SGD.
  arXiv  Detail & Related papers  (2021-03-31T02:07:18Z)
• Coded Stochastic ADMM for Decentralized Consensus Optimization with Edge Computing [113.52575069030192]
  Big data, including applications with high security requirements, are often collected and stored on multiple heterogeneous devices, such as mobile devices, drones and vehicles. Due to the limitations of communication costs and security requirements, it is of paramount importance to extract information in a decentralized manner instead of aggregating data to a fusion center. We consider the problem of learning model parameters in a multi-agent system with data locally processed via distributed edge nodes. A class of mini-batch alternating direction method of multipliers (ADMM) algorithms is explored to develop the distributed learning model.
  arXiv  Detail & Related papers  (2020-10-02T10:41:59Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Communication-Efficient Distributed Stochastic AUC Maximization with Deep Neural Networks [50.42141893913188]
  We study a distributed variable for large-scale AUC for a neural network as with a deep neural network. Our model requires a much less number of communication rounds and still a number of communication rounds in theory. Our experiments on several datasets show the effectiveness of our theory and also confirm our theory.
  arXiv  Detail & Related papers  (2020-05-05T18:08:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.