Probabilistic solution of non-linear random ship roll motion by data-driven method

•Data-Driven Approach: the article adopts a data-driven approach to study the probability density function of nonlinear random ship roll motion. This method integrates the principle of maximum entropy, dimensional analysis, pseudoinverse algorithm, and BP neural network, marking a new application in the field of studying probability density functions of ship motion.•Establishment of neural network Model: by training a BP neural network model, the article establishes a mathematical model for the steady-state probability density function of ship roll motion. This allows for the rapid acquisition of steady-state probability densities under different system characteristics and excitation intensities. The accuracy of the probability density function improves with an increase in simulated data.•Study of multistability Phenomenon: the article explores the multistability phenomenon of nonlinear random ship roll motion, providing a new perspective for understanding ship motion under complex sea conditions.•Wide Applicability: Finally, the article verifies that this data-driven approach is not only applicable to nonlinear ship roll systems but also highly feasible for other nonlinear random vibration systems and linear systems with external random excitation. In this paper, a data-driven method is employed to investigate the probability density function (PDF) of nonlinear stochastic ship roll motion. The mathematical model of ship roll motion comprises a linear term with cubic damping and a nonlinear restoring moment represented as an odd-degree polynomial up to the fifth order. The data-driven method integrates maximum entropy, the pseudo-inverse algorithm, and a backpropagation (BP) neural network to obtain the PDF. The process begins with simulating data for the nonlinear stochastic system, followed by dimensional analysis to identify dimensionless parameter clusters. Optimization algorithms are then employed to solve for the coefficients, leading to the development of a BP neural network model trained to predict the PDF across various system characteristics and excitation intensities. The method's effectiveness is validated with Monte Carlo simulations, demonstrating high accuracy and reduced sensitivity to parameter variations.