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Abstract <p>This study focuses on the application of U-Net-based neural network methods for solving the Poisson equation in a square domain. Traditional data-driven approaches using convolutional neural networks require thorough examination, particularly in scenarios with non-zero boundary conditions and arbitrary right-hand sides. The novel approach of generating datasets for training enables the use of a single neural network with a small number of parameters to efficiently obtain solutions under these varied conditions. Our experiments show that bilinear interpolation in the U-Net decoder yields better solutions compared to transposed convolution, reducing the number of trainable parameters significantly. Additionally, a hybrid method is proposed, where neural network predictions are employed to accelerate the conjugate gradient method (CG). The study demonstrates that the neural network can effectively approximate solutions for different grid sizes and achieve a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(22\%\)</EquationSource> <!--LobJMat2561012Sakharov-m1--> </InlineEquation> acceleration of CG on fine grids, thus highlighting the potential for integrating neural networks into traditional numerical solvers to enhance computational efficiency.</p>

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A Hybrid Method for Solving the Two-Dimensional Poisson Equation: Combining U-Net and Conjugate Gradient Method

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • D. I. Sakharov,
  • C. A. Tsgoev,
  • R. I. Mullyadzhanov

摘要

Abstract

This study focuses on the application of U-Net-based neural network methods for solving the Poisson equation in a square domain. Traditional data-driven approaches using convolutional neural networks require thorough examination, particularly in scenarios with non-zero boundary conditions and arbitrary right-hand sides. The novel approach of generating datasets for training enables the use of a single neural network with a small number of parameters to efficiently obtain solutions under these varied conditions. Our experiments show that bilinear interpolation in the U-Net decoder yields better solutions compared to transposed convolution, reducing the number of trainable parameters significantly. Additionally, a hybrid method is proposed, where neural network predictions are employed to accelerate the conjugate gradient method (CG). The study demonstrates that the neural network can effectively approximate solutions for different grid sizes and achieve a \(22\%\) acceleration of CG on fine grids, thus highlighting the potential for integrating neural networks into traditional numerical solvers to enhance computational efficiency.