###### Abstract
[19] proposed a method using multivariate convolutional LSTM (MC-LSTM) and mixed probabilistic principal component analysis (MPPCA) to detect anomalies in spatiotemporal data collected from space. This method was based on the LSTM model, and by adding convolutional layers, spatial and temporal features can be extracted from spatiotemporal data. [PERSON] and [PERSON] [20] proposed a short-term electricity load and price forecasting algorithm based on LSTM-Neural Network (LSTM-NN). That algorithm was based on the combination of LSTM and Neural Network, and improved the prediction performance by optimizing the parameters of LSTM and Neural Network. He and [PERSON] [21] discussed the Homotopy Perturbation Method (HPM) for nonlinear vibrations, focusing on challenges like non-conservative oscillators. It highlights key categories, provides heuristic examples, and the authors' previous work on HPM. They concentrated on the Homotopy Perturbation Method (HPM) and its applications in solving nonlinear vibration problems, including non-conservative oscillators. [PERSON] and [PERSON] [22] explored vibration phenomena in nanophysics using an efficient method based on ancient Chinese algorithms. It accurately determines amplitude-frequency relationships, offering valuable insights for nanophysics research. [PERSON] and [PERSON] [23] introduced the gamma function method to solve nonlinear cubic-quintic Duffing oscillators in one step, demonstrating its simplicity and accuracy for small amplitude oscillations. Their works seem to be more mathematically oriented, concentrating on analytical techniques to tackle nonlinear vibration problems, rather than building a database of nonlinear numerical data for machine learning. In the era of digital twins (DT), [PERSON] et al. [24] presented an AI-based fault diagnosis framework for industrial DT systems, including the triplex pump and transmission system. It combines optimization techniques with machine learning. Experimental results show FPA-CART and FPA-RF models maintain fault detection accuracy rates of 96.8% and 85.7%, outperforming other models in critical IIoT fields. [PERSON] et al. [25] introduced a digital twin-driven method for rolling bearing fault diagnosis in scenarios with limited training data. A virtual representation was created, and a Transformer-based network learned from it. Results demonstrated its effectiveness in real-world, unlabeled, and unknown health condition data. These works focused on fault diagnosis in industrial settings using digital twin-driven approaches, addressing scenarios where sufficient training data might be lacking. The present work used Duffing equation to simulate the general nonlinear vibration situation and divided it into the special case of undamped free vibration and the condition of damped forced vibration. When the Duffing equation is in undamped free vibration, with the different initial conditions, the convergence result changes between 1 or \(-1\) within a certain interval, and such changes are irregular. In addition, when the Duffing equation is in damped forced vibration, its solution will vary depending on the initial conditions, external forces, etc., and the final vibration condition will be different, and there may be multiple solutions. This study will observe the single-cycle convergent solution of the Duffing equation, especially the periodic solution of this system when the convergent trajectory is a single closed curve. In order to collect the convergence results of the Duffing equation with two different types, this study uses the fourth-order Runge-Kutta method to collect the results of two different types, and changes the initial conditions and other parameters for analysis. Then, using the supervised learning method in machine learning, the undamped free vibration type is guided by the labeled method to guide the machine to analyze the target object cognitively.