###### Abstract  
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. * [leftmargin=*] **Citation:**: [PERSON]; [PERSON] Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning. understanding of the oscillation characteristics of the Duffing equation without damping. In another study, [PERSON] [2] investigated the behavior of the Duffing equation in undamped free vibration and damped forced vibration scenarios. For undamped free vibration, [PERSON] explained that varying initial conditions lead to the convergence results switching between \(1\) and \(-1\) within a specific interval, exhibiting irregular changes. In the case of damped forced vibration, when specific initial conditions are fixed, the solution of the Duffing equation demonstrates various convergence patterns, such as single-cycle, double-cycle, and four-cycle, among others. [PERSON] et al. [3] studied the bifurcation behavior of a Duffing oscillator in damped forced vibration, and used a bifurcation diagram and Lyapunov exponent to analyze the chaotic phenomenon of the system when the external force amplitude increased. Then, the Ott-Grebogi-Yorke (OGY) control method was used to stabilize the chaotic behavior and reduce its amplitude. They demonstrated the effectiveness of the OGY control method with numerical simulations, and confirmed that the types of multiple solutions in nonlinear equations have the potential to be studied. Therefore, this research is inspired to explore the types of various solutions of the Duffing equation. Compared with general analytical or numerical analysis, since the nonlinear solution takes a lot of computing time, this research will try to use machine learning to judge the value of multiple solutions of the Duffing equation. Machine learning has experienced significant growth and has become widely utilized in various domains, leading to its rapid development in modern times. This powerful technology finds applications in diverse fields such as image recognition, search engines, industrial analysis, and artificial intelligence itself. The utilization of machine learning, along with the advancements in artificial intelligence, has shown substantial improvement in recent years, as indicated by statistics from [PERSON] [4]. The trend of artificial intelligence application has evolved over time, progressing from the use of code command systems to the impact of machine learning and the widespread adoption of neural networks in the late 1990s and early 2000s. The continuous development of sophisticated algorithms and architectures, such as the three-layered model proposed by [PERSON] et al.