<p>This paper investigates soliton solutions of the Lonngren-wave equation using a reliable analytical method known as the Hirota bilinear approach. By applying the bilinear operator, the Lonngren-wave equation is transformed into a bilinear form, which facilitates the derivation of various types of soliton solutions. These include lump waves, rogue waves, one-soliton and two-soliton solutions, as well as interaction solutions between kink-type solitary waves and rogue waves. The obtained solutions, novel in nature, exhibit a well-balanced interplay among the nonlinear physical parameters, highlighting the innovative aspect of this study. Furthermore, the Painlevé analysis is employed to verify the integrability of the Lonngren-wave equation. This analysis serves as a crucial mathematical tool for confirming the compatibility criteria of nonlinear integrable systems. Additionally, we utilize a machine learning algorithm, the Multi-Layer Perceptron regressor, to predict the performance of the acquired soliton solutions. The results are presented graphically through 2D plots, 3D surface plots, and contour graphs, offering comprehensive visualization and interpretation of the dynamics governed by individual parameter values.</p>