Analytical evaluations using neural network-based method for wave solutions of combined Kairat-II-X differential equation in fluid mechanics.

In this paper, a symbolic computation method based on a neural network architecture, the improved neural network-based method, for obtaining novel exact solutions to combined Kairat-II-X differential equation is proposed. We secure various types of soliton solutions and periodic waves through the considered approach. Furthermore, similar to existing neural network-based schemes, this improved technique also applies the output of neural networks obtained via feedforward computation as a trial function. 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 combined Kairat-II-X differential equation into a problem of determining these parameters. Using neural network-based technique and the improved variant, we derive a number of exact solutions including dark solitons, singular solitons, combined hyperbolic function solutions for Kairat-II-X equation. The proposed method is compared in detail with physics-informed neural networks in terms of computational theory. The physical relevance of the driven solutions is carefully examined by providing a range of graphs that show how the solutions behave for particular parameter values. Our findings suggest that they could be applied in the future to determine novel and diverse solutions to nonlinear evolution equations that arise in mathematical physics and engineering.