Machine Learning-Based Preconditioner to Solve Poisson Equation

Ekaterina Chekmeneva

doi:10.1007/978-3-031-97596-7_25

Machine Learning-Based Preconditioner to Solve Poisson Equation

2026-01-01

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AI

DOI：10.1007/978-3-031-97596-7_25

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In this paper, we present an attempt to construct a preconditioner based on the machine learning to solve Poisson equation. We use the Conjugate Gradient method. To precondition the algorithm we suggest approximating the inverse Laplace operator with using the U-Net. We consider the supervised learning where the vector of unknowns and right-hand sides are known; thus, we use the relative L-2 error as the loss function of the network training. We illustrate that U-Net with five convolutional layers provide insufficient accuracy of inverse Laplace operator approximation, so that the constructed conjugate gradient method stabilizes and possesses irreducible residual.

Keywords:

Poisson equation

Conjugate gradient

preconditioner

Machine Learning

Machine Learning-Based Preconditioner to Solve Poisson Equation

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