## 1 Introduction  
Weather forecasting plays a crucial role in modern society, impacting various sectors from agriculture and transportation to daily life planning [1]. Meteorological multivariate time series collected from weather stations serve as one of the primary data sources for weather forecasting tasks, containing multiple variables such as temperature, humidity, pressure, wind velocity, and precipitation which can be used to describe the weather state at a specific location over a period of time. Traditional numerical weather prediction (NWP) methods applied to multivariate time series forecasting tasks primarily consist of statistical and mathematical models, including Autoregressive Integrated Moving Average (ARIMA) [2], Kalman filtering [3], and Singular Spectrum Analysis (SSA) [4]. However, these which often struggle with nonlinear relationships and complex patterns among meteorological series [5]. Recently, the advancement of deep learning has transformed weather forecasting capabilities, driven by three key factors:algorithmic innovations, the unprecedented scale of available weather data, and advances in parallel computing hardware [6]. Deep learning time series forecasting models excel at automatically extracting complex temporal patterns and dependencies from time series data, achieving state-of-the-art accuracy [7,8]. Meteorological variables possess distinct physical characteristics that significantly influence their temporal dynamics and interrelationships. Temperature exhibits daily and seasonal cyclical patterns with gradual transitions, while precipitation demonstrates intermittent and often non-Gaussian distributions with sudden onsets and varying intensities [9]. Atmospheric pressure typically displays smoother temporal transitions, but can undergo abrupt changes during weather front passages. These variables are further governed by fundamental physical principles such as energy conservation and thermodynamic laws that impose constraints on their possible values and rates of change [10]. However, deep learning models often fail to properly account for meteorology-specific patterns of different meteorological variables such as the strict physical constraints and boundary conditions inherent in weather systems, potentially leading to imprecise or unreliable predictions. Meanwhile, due to the complexity and dynamic nature of these physical characteristics, which exhibit subtle or significant variations across different time periods and geographical regions [11], predefining these relationships in a static way would inevitably affect prediction accuracy and generalization capability. Furthermore, many advanced deep learning models employ complex architectures with massive parameter counts, resulting in high computational resource requirements. These issues limit their practical deployment in real-world timely weather forecasting systems where both accuracy and computational efficiency are critical [12]. To address these challenges, we propose a novel deep learning plug-and-play framework named Post Constraint and Correction (PCC). PCC explicitly incorporates data characteristics that emerge from complex physical processes into the normal prediction process, including numerical constraints and variable relationships, thereby contributing to prediction reliability. Our approach introduces two key sub-modules: (1) the Multi-variants Correlation Constraint (MCC) module, which captures and maintains the complex interdependencies between different meteorological variables, and (2) the State Correction (SC) module, which ensures the reasonability of predictions by incorporating additional correction terms into the final prediction. By introducing network modules that can adaptively learn the empirical relationships between meteorological variables, deep learning models can effectively capture the underlying dynamics without requiring precise definitions of basic laws.