TY - JOUR

T1 - A Hybrid Method for Solving the Two-Dimensional Poisson Equation: Combining U-Net and Conjugate Gradient Method

AU - Sakharov, D. I.

AU - Tsgoev, C. A.

AU - Mullyadzhanov, R. I.

N1 - Sakharov, D.I., Tsgoev, C.A. & Mullyadzhanov, R.I. A Hybrid Method for Solving the Two-Dimensional Poisson Equation: Combining U-Net and Conjugate Gradient Method. Lobachevskii J Math 46, 3777–3790 (2025). https://doi.org/10.1134/S1995080225610124 This research was supported by the Russian Science Foundation (grant no. 19-79-30075-П). The numerical tools were developed under a state contract with IT SB RAS (FWNS-2022-0009).

PY - 2025

Y1 - 2025

N2 - 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 acceleration of CG on fine grids, thus highlighting the potential for integrating neural networks into traditional numerical solvers to enhance computational efficiency.

AB - 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 acceleration of CG on fine grids, thus highlighting the potential for integrating neural networks into traditional numerical solvers to enhance computational efficiency.

KW - Poisson equation

KW - data-driven modeling

KW - hybrid method

KW - machine learning

KW - HYBRID METHOD

KW - MACHINE LEARNING

KW - POISSON EQUATION

KW - DATA-DRIVEN MODELING

UR - https://www.mendeley.com/catalogue/5834bc86-fab4-3fd9-bffd-f4382875254c/

UR - https://elibrary.ru/item.asp?id=88772139

UR - https://elibrary.ru/item.asp?id=88772139

U2 - 10.1134/S1995080225610124

DO - 10.1134/S1995080225610124

M3 - Article

VL - 46

SP - 3777

EP - 3790

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 8

ER -