The initial field has a crucial influence on numerical weather prediction (NWP). Data assimilation (DA) is a reliable method to obtain the initial field of the forecast model. At the same time, data are the carriers of information. Observational data are a concrete representation of information. DA is also the process of sorting observation data, during which entropy gradually decreases. Four-dimensional variational assimilation (4D-Var) is the most popular approach. However, due to the complexity of the physical model, the tangent linear and adjoint models, and other processes, the realization of a 4D-Var system is complicated, and the computational efficiency is expensive. Machine learning (ML) is a method of gaining simulation results by training a large amount of data. It achieves remarkable success in various applications, and operational NWP and DA are no exception. In this work, we synthesize insights and techniques from previous studies to design a pure data-driven 4D-Var implementation framework named ML-4DVAR based on the bilinear neural network (BNN). The framework replaces the traditional physical model with the BNN model for prediction. Moreover, it directly makes use of the ML model obtained from the simulation data to implement the primary process of 4D-Var, including the realization of the short-term forecast process and the tangent linear and adjoint models. We test a strong-constraint 4D-Var system with the Lorenz-96 model, and we compared the traditional 4D-Var system with ML-4DVAR. The experimental results demonstrate that the ML-4DVAR framework can achieve better assimilation results and significantly improve computational efficiency.

In recent years, the advancement of deep learning has led to the utilization of related technologies to enhance the efficiency and accuracy of scientific computing. Physics-Informed Neural Networks (PINNs) are a type of deep learning method applied to scientific computing, widely used to solve various partial differential equations (PDEs), demonstrating tremendous potential. This study improved upon original PINNs and applied them to forward and inverse problems in the nonlinear science field. In addition to incorporating the constraints of PDEs, the improved PINNs added constraints on gradient information, which further enhanced the physical constraints. Moreover, an adaptive learning method was used to update the weight coefficients of the loss function and dynamically adjust the weight proportion of each constraint term. In the experiment, the improved PINNs were used to numerically simulate localized waves and two-dimensional lid-driven cavity flow described by partial differential equations. Meanwhile, we critically evaluate the accuracy of the prediction results. Furthermore, the improved PINNs were utilized to solve the inverse problems of nonlinear PDEs, where the results showed that even with noisy data, the unknown parameters could be discovered satisfactorily. The study results indicated that the improved PINNs were significantly superior to original PINNs, with shorter training time, increased accuracy in prediction results, and greater potential for application.

Applying finite‐element vertical discretization to a mass‐based non‐hydrostatic kernel has proved difficult due to the constraints of vertical operators. This article proposes a novel hybrid finite‐element vertical discretization method for a semi‐implicit mass‐based non‐hydrostatic kernel, which integrates a finite‐differential scheme and a finite‐element scheme. In the hybrid method, the finite‐differential scheme which satisfies the set of constraints is applied to the linear part, while a cubic finite‐element scheme with high‐order accuracy is applied to the non‐linear part. Furthermore, to improve the accuracy of the linear part, an enlarged set of vertical levels is applied to the differential scheme. This set of vertical levels is only used to solve semi‐implicit equations, and has no impact on the grid point calculation and spectral transformations. A series of 2D idealized test cases are conducted to verify the stability and the accuracy of our new method.

The prediction of chaotic time series systems has remained a challenging problem in recent decades. A hybrid method using Hankel Alternative View Of Koopman (HAVOK) analysis and machine learning (HAVOK-ML) is developed to predict chaotic time series. HAVOK-ML simulates the time series by reconstructing a closed linear model so as to achieve the purpose of prediction. It decomposes chaotic dynamics into intermittently forced linear systems by HAVOK analysis and estimates the external intermittently forcing term using machine learning. The prediction performance evaluations confirm that the proposed method has superior forecasting skills compared with existing prediction methods.

High-resolution synthetic aperture radar (SAR) wind observations provide fine structural information for tropical cycles and could be assimilated into numerical weather prediction (NWP) models. However, in the conventional method assimilating the u and v components for SAR wind observations (SAR_uv), the wind direction is not a state vector and its observational error is not considered during the assimilation calculation. In this paper, an improved method for wind observation directly assimilates the SAR wind observations in the form of speed and direction (SAR_sd). This method was implemented to assimilate the sea surface wind retrieved from Sentinel-1 synthetic aperture radar (SAR) in the basic three-dimensional variational system for the Weather Research and Forecasting Model (WRF 3DVAR). Furthermore, a new quality control scheme for wind observations is also presented. Typhoon Lionrock in August 2016 is chosen as a case study to investigate and compare both assimilation methods. The experimental results show that the SAR wind observations can increase the number of the effective observations in the area of a typhoon and have a positive impact on the assimilation analysis. The numerical forecast results for this case show better results for the SAR_sd method than for the SAR_uv method. The SAR_sd method looks very promising for winds assimilation under typhoon conditions, but more cases need to be considered to draw final conclusions.

A coupled method of Laplace transform and Legendre wavelets is presented to obtain exact solutions of Lane‐Emden‐type equations. By employing properties of Laplace transform, a new operator is first introduced and then its Legendre wavelets operational matrix is derived to convert the Lane‐Emden equations into a system of algebraic equations. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The results show that the proposed method is very effective and easy to implement.