A new numerical method based on physics-informed neural networks for solving an optimal control problem governed by scale-invariant heat transfer equation
DOI:
https://doi.org/10.24193/subbmath.2026.3.9Keywords:
Optimal control problems, scale-invariant one-dimensional heat transfer equation, Pontryagin’s minimum principle, physics-informed neural networks.Abstract
In this paper, we propose a novel numerical method based on physics informed neural networks (PINNs) for solving a boundary optimal control problem governed by a scale-invariant one-dimensional heat equation with two-point boundary conditions. The original infinite-dimensional problem is first transformed into a finite-dimensional optimal control problem by combining finite difference discretization in space with a trapezoidal approximation of the cost functional. Next, Pontryagin’s minimum principle is applied to derive the necessary optimality conditions, leading to a coupled Hamilton–Pontryagin system formulated as a nonlinear two-point boundary value problem (TPBVP). This TPBVP is then solved using a PINNs approach, which enables the simultaneous approximation of the state and adjoint variables while naturally enforcing the boundary and optimality conditions through a unified loss function. Numerical experiments demonstrate the effectiveness of the proposed method in accurately approximating the optimal scale-invariant boundary controls. The results are validated through comparison with classical numerical methods, showing that the proposed approach achieves competitive accuracy while offering greater flexibility for handling complex problems.
Mathematics Subject Classification (2010): 49K25, 65K10.
Received 03 March 2026; Accepted 08 July 2026.
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