The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task and, since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

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.

Cloud detection is an important and difficult task in the pre-processing of satellite remote sensing data. The results of traditional cloud detection methods are often unsatisfactory in complex environments or the presence of various noise disturbances. With the rapid development of artificial intelligence technology, deep learning methods have achieved great success in many fields such as image processing, speech recognition, autonomous driving, etc. This study proposes a deep learning model suitable for cloud detection, Cloud-AttU, which is based on a U-Net network and incorporates an attention mechanism. The Cloud-AttU model adopts the symmetric Encoder-Decoder structure, which achieves the fusion of high-level features and low-level features through the skip-connection operation, making the output results contain richer multi-scale information. This symmetrical network structure is concise and stable, significantly enhancing the effect of image segmentation. Based on the characteristics of cloud detection, the model is improved by introducing an attention mechanism that allows model to learn more effective features and distinguish between cloud and non-cloud pixels more accurately. The experimental results show that the method proposed in this paper has a significant accuracy advantage over the traditional cloud detection method. The proposed method is also able to achieve great results in the presence of snow/ice disturbance and other bright non-cloud objects, with strong resistance to disturbance. The Cloud-AttU model proposed in this study has achieved excellent results in the cloud detection tasks, indicating that this symmetric network architecture has great potential for application in satellite image processing and deserves further research.

A general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

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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.

A comparative study is presented about the Adomian’s decomposition method (ADM), variational iteration method (VIM), and fractional variational iteration method (FVIM) in dealing with fractional partial differential equations (FPDEs). The study outlines the significant features of the ADM and FVIM methods. It is found that FVIM is identical to ADM in certain scenarios. Numerical results from three examples demonstrate that FVIM has similar efficiency, convenience, and accuracy like ADM. Moreover, the approximate series are also part of the exact solution while not requiring the evaluation of the Adomian’s polynomials.

This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.