This paper extends the (2+1)-dimensional Eckhaus-type dispersive long wave
equations in continuous medium to their fractional partner, which is a model
of nonlinear waves in fractal porous media. The derivation is shown briefly
using He?s fractional derivative. Using the semi-inverse method, the
variational principles are established for the fractional system, which up
to now are not discovered. The obtained fractal variational principles are
proved correct by minimizing the functionals with the calculus of
variations, and might find potential applications in numerical modelling.
El Niño is an important quasi-cyclical climate phenomenon that can have a significant impact on ecosystems and societies. Due to the chaotic nature of the atmosphere and ocean systems, traditional methods (such as statistical methods) are difficult to provide accurate El Niño index predictions. The latest research shows that Ensemble Empirical Mode Decomposition (EEMD) is suitable for analyzing non-linear and non-stationary signal sequences, Convolutional Neural Network (CNN) is good at local feature extraction, and Recurrent Neural Network (RNN) can capture the overall information of the sequence. As a special RNN, Long Short-Term Memory (LSTM) has significant advantages in processing and predicting long, complex time series. In this paper, to predict the El Niño index more accurately, we propose a new hybrid neural network model, EEMD-CNN-LSTM, which combines EEMD, CNN, and LSTM. In this hybrid model, the original El Niño index sequence is first decomposed into several Intrinsic Mode Functions (IMFs) using the EEMD method. Next, we filter the IMFs by setting a threshold, and we use the filtered IMFs to reconstruct the new El Niño data. The reconstructed time series then serves as input data for CNN and LSTM. The above data preprocessing method, which first decomposes the time series and then reconstructs the time series, uses the idea of symmetry. With this symmetric operation, we extract valid information about the time series and then make predictions based on the reconstructed time series. To evaluate the performance of the EEMD-CNN-LSTM model, the proposed model is compared with four methods including the traditional statistical model, machine learning model, and other deep neural network models. The experimental results show that the prediction results of EEMD-CNN-LSTM are not only more accurate but also more stable and reliable than the general neural network model.
Chaotic systems are complex dynamical systems that play a very important role in the study of the atmosphere, aerospace engineering, finance, etc. To improve the accuracy of chaotic time series prediction, this study proposes a hybrid model CEEMDAN-LSTM which combines Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and long short-term memory (LSTM). In the model, the original time series is decomposed into several intrinsic mode functions (IMFs) and a residual component. To reduce the difficulty of predicting chaotic time series and provide a high level of predictive accuracy, the LSTM prediction model is built for all each characteristic series from CEEMDAN deposition. Finally, the final prediction results are obtained by combining all the prediction sequences. To test the effectiveness of this model we proposed, we examined the CEEMDAN-LSTM model using the Lorenz-63 system. Further compared to Autoregressive Integrated Moving Average (ARIMA ), Support Vector Regression (SVR), multilayer perceptron (MLP), and the single LSTM model, the results of the experiment show that the proposed model performs better in the prediction of chaotic time series. Besides, the hybrid model proposed in this paper has better results than the LSTM model alone. Therefore, hybrid models based on deep learning methods and signal decomposition methods have great potential in the field of chaotic time series prediction.
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