In this paper, a mathematical model for the rolling motion of ships in random beam seas has been investigated. The ships’ steady-state rolling motion with a nonlinear restoring moment and damping effect is modeled by the nonlinear second-order differential equation. Furthermore, an artificial neural network (NN)-based, backpropagated Levenberg-Marquardt (LM) algorithm is utilized to interpret a numerical solution for the roll angle (x(t)), velocity (x′(t)), and acceleration (x′′(t)) of the ship in random beam seas. A reference data set based on numerical examples of the mathematical model for a rolling ship for the LM-NN algorithm is generated by the numerical solver Runge–Kutta method of order 4 (RK-4). The LM-NN algorithm further uses the created data set for the validation, testing, and training of approximate solutions. The outcomes of the design paradigm are compared with those of the homotopy perturbation method (HPM), optimal homotopy analysis method (OHAM), and RK-4. Statistical analyses of the mean square error (MSE), regression, error histograms, proportional performance, and computational complexity further validate the worth of the LM-NN algorithm.