## 5 Deep Learning Prediction Results and Model Comparison  
### Duffing Equation Undamped Free Vibration Type  
Before making predictions, it is necessary to evaluate the performance of the model being used. We assess the training time using both a CPU and GPU and compare the results. There are significant benefits to using GPU (Graphics Processing Unit) computing over CPU (Central Processing Unit) computing in machine learning: (1) Parallel processing: GPUs are designed to handle thousands of parallel tasks simultaneously, while CPUs are optimized for sequential processing. In machine learning, tasks like matrix multiplication and deep neural network training involve extensive parallel computations. GPUs can perform these tasks much faster due to their parallel architecture. (2) Speed: The parallelism of GPUs leads to substantially faster training and inference times for machine learning models. What might take days or weeks on a CPU can often be accomplished in hours or minutes on a GPU. This speedup is crucial for tasks like hyperparameter tuning and large-scale model training. (3) Large data: Machine learning often involves working with large datasets. GPUs have larger memory bandwidth, which allows them to efficiently handle and process large volumes of data. This makes them well-suited for tasks like image recognition and natural language processing. (4) Deep learning: Deep learning, a subset of machine learning, relies heavily on neural networks with numerous layers and parameters. Training Deep  
Figure 33: LSTM-NN model architecture for final use. Figure 32: The loss of LSTM-NN with different numbers of neurons. Neural Networks is computationally intensive, and GPUs excel in this area. They have become a standard tool for deep learning practitioners. Python provides several libraries and frameworks that enable GPU computing for users. Some of the most commonly used libraries for GPU-accelerated computing in Python include: (1) CUDA: CUDA (Compute Unified Device Architecture) is a parallel computing platform and API created by NVIDIA for GPU programming. It provides a set of tools and libraries for GPU computing and is widely used in conjunction with NVIDIA GPUs. (2) CuPy: CuPy is a GPU-accelerated library that is similar to NumPy, a popular library for numerical operations in Python. CuPy allows you to perform array operations on the GPU, making it suitable for tasks like deep learning. (3) TensorFlow: TensorFlow, a popular deep learning framework, has built-in GPU support. It can automatically offload computations to the GPU when available, significantly speeding up training and inference for Deep Neural Networks. In this study, TensorFlow was used for GPU computing. Accessing it is not challenging for researchers or practitioners. We would like to clarify our current hardware setup, which comprises an 11 th Gen Intel Core i9-11900 CPU and an NVIDIA GeForce RTX 2080 SUPER GPU. While we acknowledge that this hardware configuration may not represent the most standard or widely available setup, it is worth noting that even with this hardware, our GPU-based training demonstrated impressive computational speed, as evidenced in Figures 34 and 35. The notable speed advantage provided by the GPU is indeed a testament to the efficiency of modern hardware for machine learning tasks, especially when harnessed through optimized software, such as Python, as used in our study. This aligns with our intention to make the most of available resources and software capabilities. Figure 34 illustrates the training time for the first five epochs when using a CPU, while Figure 35 shows the training time when using a GPU for the same model. From Figures 34 and 35, it is evident that the GPU significantly outperforms the CPU in terms of training time. This discrepancy can be attributed to the GPU's superior computing power and memory bandwidth, enabling more efficient parallel computing and faster model training. As a result, GPUs are well-suited for training deep learning models.