## 2 Theoretical Analysis  
This study focuses on the Duffing equation, a representative nonlinear second-order differential equation commonly employed to simulate general nonlinear excited vibration systems. The equation describes various nonlinear vibration phenomena applicable to aircraft wing and automobile vibrations. Specifically, this research delves into the un damped free vibration type of the Duffing equation. When the linear and nonlinear elastic coefficients are identical, the solution converges to either \(+1\) or \(-1\). On the other hand, for the damped forced vibration type, the solution exhibits chaotic behavior in certain cases, while others converge to a limit cycle oscillation (LCO) result. For this situation, there is no analytical solution available so far. To address these phenomena, we numerically collect results for the \(+1\) or \(-1\) solutions in the undamped free vibration of the Duffing equation, along with the maximum and minimum displacements and velocities in the limit cycle oscillation of the damped forced vibration. Subsequently, we utilize deep learning methods to predict multiple solutions. In undamped free vibration of the Duffing equation, the convergence alternates between \(1\) and \(-1\) based on the initial conditions. Conversely, in the presence of damping and external forces, the Duffing equation exhibits varying solutions, resulting in multiple outcomes. These convergent solutions lack regularity and defy analytical resolution. While numerical methods can approximate solutions, this irregularity presents an ideal testing ground for machine learning, specifically assessing the accuracy of the commonly employed LSTM method in addressing such complex scenarios.  
The Duffing equation, a nonlinear second-order differential equation, finds applications in various fields, including: mechanical engineering, electrical engineering, aerospace engineering and materials science. The purpose of this article is to explore the accuracy of machine learning in analyzing classifying solutions of \(\pm 1\) and limit cycle oscillation of the Duffing equation. This article focuses on two main themes: undamped free vibration and damped forced vibration. The characteristics of the undamped case include classifying solutions of \(\pm 1\), which can be applied in mechanical engineering for some simple vibrations, among other applications. The damped system, on the other hand, finds use in general civil and aerospace structural vibrations. This article does not consider multi-periodic solutions or chaos due to the limit cycle oscillation is already within the practical scope of engineering component design. Beyond this range, it is not typically applicable to general engineering design. Therefore, by considering these two scenarios, this article already covers most practical engineering applications. The Duffing equation, while widely employed, poses challenges due to its nonlinear nature, often lacking analytical solutions. This study aims to investigate two specific instances of the Duffing equation: one involves classifying solutions of \(\pm 1\), and the other concerns the quadruple solutions of limit cycle oscillation. These scenarios represent undamped and damped vibration conditions, holding significant practical utility.