Phase Change Materials (PCMs) offer significant benefits for applications such as electronic cooling and Thermal Energy Storage (TES) due to their high energy storage capacity and stability. However, their limited thermal conductivity restricts widespread utilization. To address this limitation, the use of fins has become a common technique to enhance heat transfer. Optimizing fin lengths and locations is challenging, as they affect natural convection. This study aims to develop a model using a feed-forward back-propagation network, trained on simulation data, for designing a TES unit. A design matrix, incorporating factors such as fin lengths and heights, is obtained through response surface methodology. Single- and multi-objective Genetic Algorithms (GAs) are employed to seek optimal solutions, considering Complete Melting Time (CMT) and total fin length. The results demonstrate the excellent performance of the network model, with an error of less than 2.98%. The single-objective GA yields a minimum CMT. In a finless enclosure, the CMT is 338.5% higher compared to the single-objective configuration. The multi-objective GA, using Euclidean distance specified from the Pareto front to balance weight and material costs, results in a CMT that is 192% higher compared to the single-objective optimization but with a total fin length that is 286% shorter. While these improvements are significant, it is important to note that fins can increase material and manufacturing costs, as well as the overall unit weight. This artificial intelligence-computational fluid dynamics method not only predicts TES unit performance but also reduces computational costs in designing an optimal TES unit.