## 6 Conclusions  
This study investigates the convergence results of the Duffing equation in two scenarios: undamped free vibration and damped forced vibration. The RK-4 method is utilized to collect data and mark the convergence results for undamped free vibration. Subsequently, the Long Short-Term Memory (LSTM) and LSTM-Neural Network (LSTM-NN) methods are employed for training and prediction to identify the best results. Moving on, the study examines the convergence results of the Duffing equation for damped forced vibration, considering various convergence characteristics, particularly focusing on the range of single-cycle convergence results. The collected data are compiled into a dataset, and the LSTM and LSTM-NN models are established to train and predict multiple solutions for the Duffing equation with damped forced vibration. From this study, several conclusions are drawn:  
Figure 44: The training time of the first five Epochs of the LSTM model. Figure 45: The training time of the first five Epochs of the LSTM-NN model. When training the LSTM model, careful consideration must be given to the number of hidden layers, neurons, and epochs. While increasing these parameters may lead to improvements on the training set, there is a risk of overfitting on the validation set. Therefore, selecting the optimal number of layers, neurons, and epochs is essential to ensure efficient training and optimal performance on both the training and validation sets. 2. To address the issue of over-reliance on training data and mitigate overfitting, incorporating a dropout layer into the basic model structure is recommended. The dropout layer enhances training stability by preventing excessive dependence on specific neurons, reducing the risk of overfitting. 3. When selecting computing hardware for LSTM model training, using a GPU can significantly reduce the training time. GPUs are equipped with more cores and higher memory bandwidth, facilitating parallel execution of multiple matrix operations. This characteristic makes GPUs highly suitable for training deep learning models, efficiently handling the extensive computations involved in these models. Consequently, opting for GPU acceleration can lead to substantial time savings during LSTM model training. 4. In this study, LSTM models were employed to train a specific dataset of undamped and unforced Duffing equation convergence results. The achieved accuracy was 95%, although it exhibited some fluctuations. To address this, neural network hidden layers were incorporated into the LSTM model, creating the LSTM-NN model. This modification improved the accuracy to 96% and mitigated the issue of accuracy fluctuations. However, it is worth noting that the increased number of hidden layers resulted in longer training times. 5. In this study, the LSTM model was utilized to train a dataset containing convergence results of the Duffing equation with damping and external forces, focusing on single-cycle solutions. The accuracy achieved was 97.5% with fast training speed. By incorporating hidden layers of neurons (LSTM-NN), the training accuracy increased to 98% with lower loss. However, it is important to acknowledge that training time was prolonged due to the inclusion of a larger number of hidden layers. 6. The LSTM model established in this study can predict the special case of the convergence result of the Duffing equation without damping and no external force. When the predicted value is \(+1\), the accuracy reaches 96%, and when the value is \(-1\), the accuracy also reaches 86%. The model with a hidden layer of neurons was 87% accurate on \(+1\) predictions and 98% accurate on \(-1\) predictions. The LSTM model after adding the hidden layer of neurons (LSTM-NN) is more suitable for predicting binary data sets. 7.