High-dimensional meteorological data offer a comprehensive overview of meteorological conditions. Nevertheless, predicting regional high-dimensional meteorological data poses challenges due to the vast scale and rapid changes. Apart from slow conventional numerical weather prediction methods, recently developed deep learning methods often fail to fully integrate spatial information of the high-dimensional data and require a significant amount of computational resources. This paper presents the spatiotemporal analysis fitting prediction algorithm (SA-Fit), an approximation algorithm for regional high-dimensional meteorological data prediction. SA-Fit proposes two key designs to achieve efficient prediction of the high-dimensional data. SA-Fit introduces a lightweight Transformer-based spatiotemporal analysis network to encode spatiotemporal information, which can integrate the interaction information between different coordinates in the data. Furthermore, SA-Fit introduces explicit functions with a lasso penalty to fit variations in high-dimensional meteorological data, achieving the prediction of a large amount of data with minimal prediction values. We performed experiments using the ERA5 dataset from the Shanghai and Xi’an regions. The experimental results show that SA-Fit is comparable to other advanced deep learning prediction methods in overall prediction performance. SA-Fit shortens training time and significantly reduces model parameters while using the Transformer structure to ensure prediction accuracy.