||This study addresses the problem of predicting convergence outcomes in the Duffing
equation, a nonlinear second-order differential equation. The Duffing equation exhibits intriguing
behavior in both undamped free vibration and forced vibration with damping, making it a subject
of significant interest. In undamped free vibration, the convergence result oscillates randomly between
1 and −1, contingent upon initial conditions. For forced vibration with damping, multiple
variables, including initial conditions and external forces, influence the vibration patterns, leading
to diverse outcomes. To tackle this complex problem, we employ the fourth-order Runge–Kutta
method to gather convergence results for both scenarios. Our approach leverages machine learning
techniques, specifically the Long Short-Term Memory (LSTM) model and the LSTM-Neural Network
(LSTM-NN) hybrid model. The LSTM-NN model, featuring additional hidden layers of neurons,
offers enhanced predictive capabilities, achieving an impressive 98% accuracy on binary datasets.
However, when predicting multiple solutions, the traditional LSTM method excels. The research
encompasses three critical stages: data preprocessing, model training, and verification. Our
findings demonstrate that while the LSTM-NN model performs exceptionally well in predicting
binary outcomes, the LSTM model surpasses it in predicting multiple solutions.