Traditional numerical methods face a trade-off between computational cost and accuracy when solving partial differential equations. Low-order solvers are fast but less accurate, while high-order solvers are accurate but much slower. To address this challenge, we propose a novel framework, the hybrid multigrid neural operator (HMgNO). The HMgNO couples a low-order numerical solver with a multigrid neural operator, and the neural operator is used to correct the low-order numerical solutions to obtain high-order accuracy at each fixed time step size. Thus, the HMgNO achieves accurate solutions while ensuring computational efficiency. Moreover, our framework supports multiple types of low-order numerical solvers, such as finite difference and spectral methods. Experiments on the Navier-Stokes, shallow-water, and diffusion-reaction equations demonstrate that the proposed framework achieves the lowest relative error and smallest spectral bias with few model parameters and fast inference speed.

Quantitative precipitation nowcasting (QPN) is a highly challenging task in weather forecasting. The ability to provide precise, immediate, and detailed QPN products is necessary for a variety of situations, including storm warnings, air travel, and large gatherings. To address this challenge, this article proposes a new transformer lightweight physics-informed transformer (LPT)-QPN for QPN tasks, utilizing vertical cumulative liquid water content (VIL) products. This model adopts novel transformer modules to model the long-term evolution of precipitation and incorporates multihead squared attention (MHSA) to model its highly nonlinear relationships while reducing computational complexity. The results of experimental evaluations demonstrate the superiority of LPT-QPN when compared to existing state-of-the-art QPN models. In particular, the LPT-QPN model demonstrates greater accuracy for long lead time and in high-intensity areas, confirmed in both quantitative and qualitative evaluations. In addition, through three customized fine-tuning schemes, we are able to further improve the predictability of the LPT-QPN model for specific precipitation events. By incorporating the physical constraints of the convection-diffusion equation, our approach offers novel perspectives for future explorations that combine physical prior knowledge and deep-learning (DL) techniques.

Several ensemble-based three-dimensional variational (3D-Var) filters are compared. These schemes replace the static background error covariance of the traditional 3D-Var with the ensemble forecast error covariance, but generate analysis ensemble anomalies (perturbations) in different ways. However, it is demonstrated in this paper that they are all theoretically equivalent to the ensemble transformation Kalman filter (ETKF). Furthermore, a new method named EnPSAS is presented. The analysis shows that EnPSAS has a small condition number and can apply covariance localization more easily than other ensemble-based 3D-Var methods.