Stable Numerical Solution to Multi-dimensional Nonlinear Inverse Heat Conduction Problems via Artificial Neural Networks

Stable Numerical Solution to Multi-dimensional Nonlinear Inverse Heat Conduction Problems via Artificial Neural Networks Full article

Journal

Lobachevskii Journal of Mathematics
ISSN: 1995-0802 , E-ISSN: 1818-9962

Output data

Year: 2026, Volume: 47, Number: 3, Pages: 1213-1232 Pages count : 20 DOI: 10.1134/s1995080225614705

Tags

inverse heat conduction, multi-dimensional nonlinear parabolic equation, ill-posed problems, regularization, L-curve method, Beck’s future time approach, artificial neural networks (ANNs)

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 of the Russian Academy of Sciences, Novosibirsk, Russia
4	Department of Mathematics and Statistics, University of North Carolina at Charlotte, 28223, Charlotte, NC, USA

Funding (1)

Russian Science Foundation

24-41-04004

Multi-dimensional nonlinear inverse heat conduction problems are 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 either the discrepancy principle or the L-curve method. The well-known Beck future time approach is applied to improve the reconstruction of the boundary values near the final time moment. Numerical simulations show that the approach is efficient for these severely ill-posed, multi-dimensional and nonlinear problems.

Hào D.N. , Shishlenin M. , Van C.B.
Stable Numerical Solution to Multi-dimensional Nonlinear Inverse Heat Conduction Problems via Artificial Neural Networks
Lobachevskii Journal of Mathematics. 2026. V.47. N3. P.1213-1232. DOI: 10.1134/s1995080225614705 WOS Scopus OpenAlex

Submitted:	Nov 12, 2025
Accepted:	Jan 12, 2026
Published print:	Jul 6, 2026
Published online:	Jul 6, 2026

≡ Web of science:	WOS:001812865200003
≡ Scopus:	2-s2.0-105043923079
≡ OpenAlex:	W7167509522