## 5 Conclusions and Future Work  
In this paper, we have presented a generic plug-and-play approach named Post Constraint and Correction (PCC) for improving deep learning-based multivariate weather time series forecasting by explicitly incorporating domain knowledge such as the physical characteristics of meteorological variables. Our method addresses two key challenges in weather time series prediction: (1) missing or improper alignment of the variables' physical interdependencies, and (2) imprecise or unreasonable predictions caused by neural network randomness and the strict physical constraints of meteorological variables.  
Extensive experimental results across different cases clearly demonstrate the effectiveness of our PCC approach in terms of improved accuracy and cost-efficiency. For instance, our method reduces the prediction MSE by up to 11% on average with the SegRNN backbone while requiring less than 0.5% additional GPU memory consumption. Our analysis of PCC's scalability demonstrates that the proposed module enables backbone models to achieve better performance with a smaller number of parameters, making it particularly suitable for resource-constrained environments where computational efficiency is as critical as forecasting accuracy.  
Additionally, our ablation studies validate the importance of each component in the proposed method along with the reasonableness of our design. Visualization analysis further reveals the interpretability of the module's behavior, showing how it effectively learns and leverages the specific properties of meteorological time series to achieve improved model performance.  
While our PCC module demonstrates significant performance improvements, we acknowledge interpretability limitations common to deep learning approaches in physical systems. Unlike traditional physics-based models, neural networks often function as 'black boxes', making it difficult to fully explain how they incorporate domain knowledge [50]. In our opinion, there is a promising approach to address this challenge by developing hybrid architectures that more explicitly incorporate physical relationships. For example, implementing gating mechanisms that selectively apply predefined physical constraints to the model outputs may offer an optimal balance between purely data-driven approaches and physics-constrained models. Such hybrid approaches can not only enhance interpretability but might further reduce model complexity while maintaining or improving prediction accuracy. We consider this a promising direction for our ongoing research, and plan to thoroughly investigate such mechanisms in subsequent work.  
Conceptualization, Z.W., Z.L. and Y.L.; methodology, Z.W.; software, Z.W.; validation, Z.W.; formal analysis, Z.W.; investigation, Z.W. and Z.Y.; resources, Z.W. and Z.L.; data curation, Z.W. and Y.L.; writing--original draft preparation, Z.W.; writing--review and editing, Z.W., Z.L. and Z.Y.; visualization, Z.W. and Z.Y.; supervision, Z.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The dataset we used in our experiment was obtained from the open time series library at [[https://github.com/thuml/Time-Series-Library](https://github.com/thuml/Time-Series-Library)]([https://github.com/thuml/Time-Series-Library](https://github.com/thuml/Time-Series-Library)) (accessed on 29 May 2024). This dataset is preprocessed in a tabular format. The raw CSV files for different years are available at [[https://www.bgc-jena.mpg.de/wetter/weather_data.html](https://www.bgc-jena.mpg.de/wetter/weather_data.html)]([https://www.bgc-jena.mpg.de/wetter/weather_data.html](https://www.bgc-jena.mpg.de/wetter/weather_data.html)) (accessed on 29 May 2024).  
The authors declare no conflicts of interest.