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The Artificial Neural Network (ANN) has become an attractive approach in Machine Learning (ML) to analyze a complex data-driven problem. Due to its time efficient findings, it has became popular in many scientific fields such as physics, optics, and material science. This paper presents a new approach to design and optimize the electromagnetic plasmonic nanostructures using a computationally efficient method based on the ANN. In this work, the nanostructures have been simulated by using a Finite Element Method (FEM), then Artificial Intelligence (AI) is used for making predictions of associated sensitivity (S), Full Width Half Maximum (FWHM), Figure of Merit (FOM), and Plasmonic Wavelength (PW) for different paired nanostructures. At first, the computational model is developed by using a Finite Element Method (FEM) to prepare the dataset. The input parameters were considered as the Major axis, a, the Minor axis, b, and the separation gap, g, which have been used to calculate the corresponding sensitivity (nm/RIU), FWHM (nm), FOM, and plasmonic wavelength (nm) to prepare the dataset. Secondly, the neural network has been designed where the number of hidden layers and neurons were optimized as part of a comprehensive analysis to improve the efficiency of ML model. After successfully optimizing the neural network, this model is used to make predictions for specific inputs and its corresponding outputs. This article also compares the error between the predicted and simulated results. This approach outperforms the direct numerical simulation methods for predicting output for various input device parameters.