<p>The (2+1)-dimensional nonlinear dynamical equation is under investigation in this study. The considered equation models nonlinear waves such as shallow water waves and high-amplitude sound waves. To obtain a variety of exact solutions, this work employs the Riccati sub-equation neural network technique. The neural networks, which are multi-layer computational representations, consist of weights and activation functions that connect neurons across the input, hidden, and output layers. In this approach, the solution corresponding to the Riccati equation is assigned to every neuron in the first hidden layer, thereby constructing novel trial functions. Our analysis of the given equation leads to the development of generalized exact solutions expressed in terms of hyperbolic functions, trigonometric functions, rational functions, as well as bright, dark, combo, and complex solitons. These solutions validate the mathematical framework of the proposed method. Solitary waves hold significant importance in fluid mechanics as stable, localized wave packets that propagate over long distances without dispersing or changing shape. Such waves model several critical real-world phenomena, including tsunami propagation in shallow waters, internal waves that transport energy and nutrients across ocean stratifications, atmospheric waves, and tidal bores. These phenomena are governed by nonlinear dynamics, where the underlying equations include nonlinear terms that give rise to complex behaviors such as stability, energy transfer, and wave interactions. Therefore, the study of solitary waves is essential for advancing the understanding of nonlinear dynamics in fluid systems. Furthermore, through various graphical representations, the dynamic characteristics of the obtained solution sets, including wave structures, have been illustrated. The results demonstrate the effectiveness of the proposed approach, which could contribute to a deeper understanding of nonlinear dynamics in complex physical systems.</p>