Stable Numerical Solution to the Cauchy Problem for Multi-Dimensional Nonlinear Elliptic Equations via Artificial Neural Networks

Stable Numerical Solution to the Cauchy Problem for Multi-Dimensional Nonlinear Elliptic Equations via Artificial Neural Networks Full article

Journal

Computational Mathematics and Mathematical Physics
ISSN: 0965-5425 , E-ISSN: 1555-6662

Output data

Year: 2026, Volume: 66, Number: 7, Pages: 1397–1421 Pages count : 25 DOI: 10.1134/S0965542526700727

Tags

Cauchy problem, multi-dimensional nonlinear elliptic equation, ill-posed problems, regularization, L-curve method, artificial neural networks

Authors

Hào Dinh Nho ¹ , Shishlenin Maxim ^2,3 , Van Cong B. ⁴

Affiliations

1	Hanoi Institute of Mathematics, VAST, Hanoi, Vietnam
2	Sobolev Institute of Mathematics
3	Institute of Computational Mathematics and Mathematical Geophysics
4	Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223 USA

Funding (1)

Russian Science Foundation

24-41-04004

The Cauchy problem for multi-dimensional nonlinear elliptic equations is reformulated as a least squares problem and then stabilized by Tikhonov regularization method. The last is solved in the framework of artificial neural networks, while the regularization parameter is chosen by the discrepancy principle or the L-curve method. Numerical simulations show that the approach is efficient for this severely ill-posed, multi-dimensional and nonlinear problem.

Hào D.N. , Shishlenin M. , Van C.B.
Stable Numerical Solution to the Cauchy Problem for Multi-Dimensional Nonlinear Elliptic Equations via Artificial Neural Networks
Computational Mathematics and Mathematical Physics. 2026. V.66. N7. P.1397–1421. DOI: 10.1134/S0965542526700727 WOS Scopus OpenAlex

Submitted:	Sep 10, 2025
Accepted:	Apr 10, 2026
Published print:	Aug 27, 2026
Published online:	Aug 27, 2026

≡ Web of science:	WOS:001862007100009
≡ Scopus:	2-s2.0-105048646548
≡ OpenAlex:	W7204435815