E. A. Chekmeneva¹, T. S. Khachkova², V. V. Lisitsa²

¹Novosibirsk State University (Novosibirsk, Russian Federation)

²Sobolev Institute of Mathematics (Novosibirsk, Russian Federation)

Abstract. In this paper, we present a method of constructing a preconditioner based on machine learning methods for use in the numerical solution of the Poisson equation in porous media modelling. The iterative conjugate gradient method is used to solve the problem. To precondition a system of linear algebraic equations, we propose to approximate the inverse Laplace operator using a convolutional neural network of the U-net architecture. We consider two alternative approaches to the formation of a training dataset for a neural network. The first method is based on the use of pairs of vectors and on the results of applying the Laplace operator to them. In the second method, the training pairs are residual vectors obtained by implementing the conjugate gradient method and results of applying the Laplace operator to them. The neural network learning process is based on minimizing the $L^2$-norm relative error. We show that while using each of the presented learning methods, the U-net neural network with five convolutional layers approximating the inverse Laplace operator provides insufficient accuracy to reduce the number of iterations in the conjugate gradient method. Therefore, the modified conjugate gradient method is stabilized and has an irreducible residual.

Key Words: Poisson equation, method of conjugate gradients, preconditioner, machine learning

For citation: E. A. Chekmeneva, T. S. Khachkova, V. V. Lisitsa. Machine learning-based preconditioner to solve Poisson equation. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:2(2026), 83–98. DOI: https://doi.org/10.15507/2079-6900.28.202602.83-98

Information about the authors:

Ekaterina A. Chekmeneva, student at Novosibirsk State University (2 Pirogova Street, Novosibirsk, 630090, Russia), ORCID: https://orcid.org/0009-0003-8509-0534, e.chekmeneva@g.nsu.ru

Tatyana S. Khachkova, Ph.D. (Phys. and Math.), senior researcher at the Institute of Mathematics, Siberian Branch of Russian Academy of Science (4 Koptug Avenue, Novosibirsk, 630090, Russia), ORCID: https://orcid.org/0000-0002-1595-7142, hachtanya@mail.ru

Vadim V. Lisitsa, D.Sc. (Phys. and Math.), head of laboratory at the Institute of Mathematics, Siberian Branch of Russian Academy of Science (4 Koptug Avenue, Novosibirsk, 630090, Russia), ORCID: https://orcid.org/0000-0003-3544-4878, v.v.lisitsa@math.nsc.ru