## 1 Introduction  
With the development of sensor technology and the popularization of mobile Internet of Things devices, the volume of spatiotemporal data continues to increase, with the characteristics of large volume and rapid updating. These data encompass rich information across both spatial and temporal dimensions. In the spatial dimension, distribution patterns can be clustered, random, or discrete, while in the temporal dimension, the data are often monotonically increasing. These characteristics pose ongoing challenges to the organization, storage, and retrieval technologies of the system [1]. Index technology has been widely used in the management and retrieval of multi-dimensional spatial data, such as R-tree [2] and KD-tree [3]. However, a key issue with existing indexing technology is that the space consumption of the index structure will increase significantly as the amount of data increases. Research shows that under typical OLTP workloads, the index structure can occupy up to 55% of server memory [4]. Traditional indexes face challenges related to query time and space efficiency. R-treesexhibit diminishing efficiency as the dimensionality of the data increases, primarily due to the increased overlap between bounding rectangles. This overlap leads to a greater number of nodes being examined during query processing, thereby substantially increasing query time. Additionally, R-trees require the storage of multiple bounding rectangles and pointers to child nodes. In scenarios where data distribution is uneven, excessive overlap between rectangles can occur, leading to inefficient queries and wasted space, as the tree may traverse unnecessary paths. Similarly, KD-trees encounter significant challenges when dealing with high-dimensional data, a phenomenon often referred to as the \"curse of dimensionality.\" As the number of dimensions increases, partitioning strategies result in suboptimal spatial decomposition, raising the computational cost of searches and reducing the overall efficiency of KD-trees. Furthermore, the tree structure tends to become unbalanced, resulting in a larger memory consumption and prolonged query times. Geohash encoding can compactly represent two-dimensional geospatial data as a one-dimensional string, effectively preserving spatial locality and enabling efficient spatial partitioning. By subdividing the geographic space into a grid of cells, each identified by a unique alphanumeric string, Geohash allows for the quick retrieval of nearby locations based on their shared prefixes, reducing the complexity of managing multidimensional data. Geohash can efficiently process two-dimensional geographical data, but it is difficult to process higher-dimensional data. In recent years, machine learning has achieved significant advancements in image recognition, natural language processing, recommendation systems and other fields, and has also attracted the attention of researchers in the database field. In this context, [PERSON] et al. (2018) first proposed the concept of the learned index in 2018, which opened up a new direction for machine learning in the index field. Data keys are specific values that identify and locate corresponding records in a dataset, and they typically serve as inputs for the index. The core idea of the learned index is to fit the distribution of a dataset by training models, efficiently predict the location of data keys to be queried through models, and search within the error range to finally find the exact location of data keys. Traditional index structures are general data structures. Although these structures can handle a variety of data types and distributions, they often overlook specific application scenarios and the underlying characteristics of the data. For instance, traditional multi-dimensional index structures not only consume substantial storage space but also incur high scanning costs if not appropriately chosen.