A new numerical method based on physics-informed neural networks for solving an optimal control problem governed by scale-invariant heat transfer equation

Melissa Mansouri; Karim Benalia; Brahim Oukacha; Karim Beddek

doi:10.24193/subbmath.2026.3.9

Authors

Melissa Mansouri Department of Mathematics, Faculty of Sciences,\\ Laboratory of Operational Research and Mathematics of Decision, University of Tizi Ouzou, Algeria https://orcid.org/0009-0007-4778-944X
Karim Benalia Department of Mathematical Sciences, Faculty of Hydrocarbons and Chemistry, University of Boumerdes, Boumerdes, Algeria
Brahim Oukacha Department of Mathematics, University of Tizi Ouzou, Algeria https://orcid.org/0009-0006-5172-6356
Karim Beddek Department of Automation, Faculty of Hydrocarbons and Chemistry, University M’Hamed Bougara, Algeria https://orcid.org/0000-0003-2184-9399

DOI:

https://doi.org/10.24193/subbmath.2026.3.9

Abstract

In this paper, we propose a new numerical method based on physics-informed neural networks (PINNs) for solving a boundary optimal control problem governed by a scale-invariant one-dimensional heat transfer equation with two-point boundary conditions and a scaling parameter. First, the original problem is transformed into an optimal control problem governed by a system of scale-invariant ordinary differential equations (ODEs) with two-point boundary conditions using finite-difference discretization and trapezoidal approximation. Next, Pontryagin’s minimum principle is applied to derive the necessary optimality conditions, which lead to a two-point boundary value problem (TPBVP). This TPBVP is then solved using the PINNs approach to compute the optimal scale-invariant boundary control laws. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed approach, and the obtained results are validated through comparison with those generated by an alternative numerical method.

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