This paper explores a novel approach aimed at overcoming existing challenges in the realm of local search algorithms. Our aim is to improve the decision process that takes place within a local search algorithm so as to make the best possible transitions in the neighborhood at each iteration. To improve this process, we propose to use a neural network that has the same input information as conventional local search algorithms. In this paper, which is an extension of the work presented at EvoCOP2024, we investigate different ways of representing this information so as to make the algorithm as efficient as possible but also robust to monotonic transformations of the problem objective function. To assess the efficiency of this approach, we develop an experimental setup centered around NK landscape problems, offering the flexibility to adjust problem size and ruggedness. This approach offers a promising avenue for the emergence of new local search algorithms and the improvement of their problem-solving capabilities for black-box problems. The last version of this article is published in the journal SN Computer Science (Springer).