A New Hybrid Construction of Softmax Method and Physics‐Informed Neural Network Method for (3+1)‐Dimensional Potential Yu–Toda–Sasa–Fukuyama Equation

ABSTRACT

Softmax functions play an essential function in constructing the hidden layer of machine learning. Some models, such as Physics‐Informed neural networks (PINNs), could directly use the Softmax function for building a pipeline. This paper investigates the ‐dimensional potential Yu–Toda–Sasa–Fukuyama (DPYTSF) equation using the Softmax method and Physics‐Informed neural network method. The main idea investigates nonlinear wave solutions of the considered nonlinear partial differential equation through a hybrid framework that combines the Softmax method with Physics‐Informed Neural Networks. By employing a traveling‐wave transformation, the governing equation is reduced to an ordinary differential equation, which is subsequently solved using the Softmax method. Several families of exact analytical solutions are obtained, including hyperbolic, trigonometric, exponential, and rational wave structures. To further validate the derived solutions, a PINN model is constructed using both analytical solution information and the physical constraints imposed by the governing equation. The learning performance of the PINN is evaluated through prediction accuracy, training loss, and PDE residual analyses. Numerical experiments demonstrate excellent agreement between the analytical solutions and PINN predictions, confirming the effectiveness of the proposed hybrid framework. The obtained results indicate that the integration of analytical techniques and scientific machine learning provides a reliable and efficient approach for studying nonlinear evolution equations and complex wave phenomena.