Application of bilinear neural networks method to the deflection of nonlinear waves over a Kirchhoff plate

In this paper, a symbolic computation method based on a neural network architecture demonstrating the nonlinear wave dynamics governed by the model of the deflection of nonlinear waves over a Kirchhoff plate for obtaining novel exact solutions is proposed. This model is elaborated the evolution of multidirectional wave propagation in plasma physics, optical systems, and fluid mechanics. The bilinear neural network method along with Hirota bilinear method is selected to find many exact solutions to the proposed model. This emphasizes the complex interplay between nonlinearity and dispersion; the bilinear neural network method is utilized as an important hybrid strategy that combines the symbolic precision of Hirota's bilinear formalism with the adaptive learning capability of neural architectures. To validate the effectiveness of our methodology, we apply it to the nonlinear model, a prototypical nonlinear model with significance in soliton theory and wave dynamics. We secure various types of soliton solutions and lump waves through the considered approach. By introducing various activation functions, novel trial functions are extracted. These functions incorporate the neural networks' weights and biases, in that connection transforming the solution of the model of the deflection of nonlinear waves over a Kirchhoff plate into a problem of determining these parameters. Using neural network-based technique and the improved variant, we derive a number of exact solutions for the mentioned equation, including some novel solutions. The derived results emphasize the recurrence, and coherence of nonlinear patterns in multidimensional systems, highlighting that the governed bilinear-neural hybrid framework suggests a powerful and interpretable pathway for analyzing nonlinear wave evolution and intricate soliton interactions in higher-dimensional nonlinear partial differential equations.